Constructions and families of nonbinary linear codes with covering

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 5, JULY 1999

Constructions and Families of Nonbinary Linear Codes with Covering Radius Alexander A. Davydov

Abstract—New constructions of linear nonbinary codes with covering radius R = 2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R = 2 as a “seed” these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R = 2 4 and all codimensions r 3 are obtained for all alphabets of size q with the help of the constructions described. The parameters obtained are better than those of known codes. New estimates for some partition parameters in earlier known constructions are used to design new code families. Complete caps and other saturated sets of points in projective geometry are applied as starting codes. A table of new upper bounds on the length function for q = 4; 5; 7; R = 2; and r 24 is included.

Index Terms—Complete caps, covering codes, covering radius, linear codes, nonbinary codes.

I. INTRODUCTION Constructions of linear covering codes with covering radius R = and close problems are considered, e.g., in [1]–[16], and the references therein. An introduction to linear covering codes and a general survey is given in [3]. Linear codes with R = 2, even basis q 4, and codimension r = 4 are constructed in [1, p. 104]. In [5] and [6] the construction of [1] is generalized to odd basis q 5: In [2] a construction of codes with R = 2; q = 3; r = 4t + 1; is given. Using these codes a “doubling” construction of [8, Theorem 3a], [14, Sec. 4] produces codes with R = 2; q = 3; r = 4t + 2: In a geometric perspective, linear covering codes with R = 2 are studied in [10], [11, Ch. 9], [12, Sec. 2], [15], and [16]. For arbitrary q and r , some code constructions were proposed in [4] and developed in [5], [6], and [9]. They rely on a linear code of covering radius R as a starting code and result in infinite families of linear codes with the same covering radius. These constructions can be called “q m concatenating constructions” since a parity-check matrix of a starting code is repeated q m times. In this correspondence we use, modify, and develop the approach and constructions considered in a recent paper [6]. In Section II, we survey constructions for R = 2 from [6] and propose new constructions. For all the constructions described we obtain estimates of partition parameters useful for constructing new codes by an iterative process when codes obtained in a certain step are starting codes in the next steps. The obtained estimates are better than those in [6] and help us to obtain a number of good code families. In Section III, with the help of the described constructions and the iterative process, we construct new infinite families of covering codes with R = 2 and any q 4; r 3: Parameters of the new codes are better than those of known codes. The obtained codes give new upper bounds on the length functions. We have included a table of the new 2

Manuscript received August 3, 1997; revised October 19, 1998. This work was supported in part by the International Science Foundation under Grant UAA300 and by the Russian Basic Research Foundation under Grant 9501-01331a. The material of this correspondence was presented in part at the 5th International Workshop on Algebraic and Combinatorial Coding Theory, ACCT’96, Sozopol, Bulgaria, June 1–7, 1996. The author is with the Institute for Information Transmission Problems, Russian Academy of Sciences, GSP-4, Moscow, 101447, Russia. Communicated by A. Barg, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(99)04381-3.

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upper bounds for codes with q = 4; 5; 7; R = 2; and r 24: Some results of this work were announced in [7] and [8]. Let F q be the Galois field of q elements and F 3q = F q nf0g: Denote by an [n; n 0 r ]q R code a q -ary linear code of length n, codimension r, and covering radius R: This code is an (n 0 r )dimensional subspace in the space of n-dimensional row vectors over F q : In the notation [n; n 0 r ]q R we may omit R and change n 0 r by its value. The length function l(r; R; q ) [1] is the smallest length of a q -ary linear code with codimension r and covering radius R: Let Gqr be the space of r-dimensional q -ary column vectors. All matrices and column vectors below are q -ary. An element of appearing in a q -ary matrix denotes a column vector of Gqm that Fq is a q -ary representation of this element, and vice versa, we can treat a column vector of Gqm as an element of F q : Let 0k be the all-zero matrix with k rows. Denote by 0 a zero column vector. We specify the dimensions of vectors and matrices unless they are unambiguously defined by the context. In this case we specify the number of q -ary rows in a matrix or positions in a column. Denote by fH g the set of columns of a matrix H: Let HC be a parity-check matrix of a code C and let T be the symbol of transposition. We consider linear combinations of q -ary columns only with nonzero q -ary coefficients. Fact 1: An [n; n 0 r]q R code with a parity-check matrix H = 1 1 1 f n ], where f j 2 Gqr ; j = 1; n, has covering radius R = 2 if and only if each nonzero column p of Gqr can be represented by one of following linear combinations with columns of H :

[f 1

p = 1 f j ; p 2 Gqr ; p 6= 0; f j 2 fH g; 1 2 F 3q p = 1 f j + 2f j ; p 2 Gqr ; p 6= 0; j1 6= j2 ; f j ; f j 2 fH g; 1 ; 2 2 F 3q

and, besides, there is a nonzero column p0 of

(1) (2)

Gqr that cannot be

written in the form of (1). Note that if R = 1, each nonzero column of Gqr can be written in the form of (1). We can treat a nonzero column of Gqr as a point of the projective geometry PG (r 0 1; q ): Then a parity-check matrix of an [n; n 0 r]q 2 code C is a set H of n points. A point set of a projective geometry is 1-saturated if any point of the geometry lies on a line through at least two points of this set. By Fact 1, H is a 1-saturated set. A complete cap is a 1-saturated set such that no three of its points are collinear. If C has minimum distance at least 4 then H is a complete cap. For details, see [6, pp. 2072, 2077], [11, Ch. 9], [12, Sec. 1.3], [13, Sec. 12], [15], and [16]. In Section III, we use complete caps and other 1-saturated sets as starting codes. Definition 1: Let H be a parity-check matrix of an [n; n 0 r]q 2 code and let H = [f 1 1 1 1 f n ], where f j 2 Gqr ; j = 1; n: a) A partition of the column set fH g into nonempty subsets is called a 2-partition if each nonzero column p of Gqr can be represented in the form of either (1) or (2) where in (2) columns f j and f j belong to distinct subsets. b) A subset F of a 2-partition is called a Q+ -subset if each column of F can be written in the form of (2), where columns f j and f j belong to distinct subsets Y and Z of this 2partition and it is possible that Y = F : A 2-partition is called a 2+ -partition if it has a Q+ -subset. c) A 2-partition is called a 2E -partition if each nonzero column p of Gqr can be represented in the form of (2) with columns

0018–9448/99$10.00  1999 IEEE

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f j and f j belonging to distinct subsets. A code is called an E -code if its parity-check matrix admits a 2E -partition.

this partition is a 2E -partition (see Remark 1), U is an E-code.

Remark 1: Definition 1 uses approaches of [6, Definitions 2.4, 6.1, 6.2]. The term “AL2” of [6] is changed to “E” (for “exactly”). Each nonzero column p of Gqr is equal to a linear combination of exactly two columns from a parity-check matrix of an [n; n 0 r ]q 2 E-code. By Fact 1 and Definition 1, a 2-partition of a parity-check matrix H is a 2E -partition if and only if each column of H can be written in the form of (2) with columns f j and f j belonging to distinct subsets. So, each subset of a 2E -partition is a Q+ -subset. Denote by h(H ; K) (respectively, h+ (H ; K) or hE (H ; K)) the number of subsets in a 2-partition K (respectively, 2+ -partition K or 2E -partition K) for a matrix H: A partition into one-element subsets is called trivial. By Fact 1, for a parity-check matrix of a code with covering radius 2 the trivial partition always is a 2partition. The minimal number of subsets in any 2-partition for a matrix H is denoted by h(H ): So, h(H ) = min h(H ; K): Similarly,

II. CONSTRUCTIONS

code

1 0 0

1 0 1

1 0

0 1 0

(see at the bottom of this page), where

f j 2 fH0 g; f1 ; 2 ; 1 1 1 ; q g = F q j 2 F q ;

E

0 1 1

= [f 1

f 2 f 3 f 4 f 5 ]:

(we will write partitions in this form). Assume ; ;

(3)

2 F 34 : We have

T

=

b) Assume p = 1 f j + 2 f j with j1 6= fH0 g; see (2). For j1 ; j2 6= n0 we have (p; u1 ; u2 )

T

Values of

=

1 0 0

1 0 1

1 0 2

0 1 0

0 1 1

0 1 2

= [f 1

1 1 1 f 6 ]:

HV0

r

0

=

Wm m

0

r m m

j f1 f1 1 1 1 f1 j 1 2 1 1 1 j 1 1 1 2 1 1 1 1 q

q

+ 2 (f j

; y ; j y )T ;

1 ; 2 2 F 3q : (8)

can be found from the system

(p; u 1 ; u 2 ) = 1 (f j T

= 2f 2+j + 4f 3+j ; j = 0; 3

j 0 j 0 j W

x ; y

; x ; j x )T

that has determinant 1 2 ( j 0 j ) = 6 0: Here since j1 6= j2 : For j2 = n0 we have

(4)

Using Fact 1, we verified directly that U has covering radius 2. Hence the trivial partition of HU is a 2-partition. Since

f 1+j

= 1 (f j

2

j2 ; f j ; f j

1 x + 2 y = u1

1 j x + 2 j y = u2

with parity-check matrix

HU

j = 1; n0 0 1; i 6= j

i 6= j; 0 is the zero column with m positions, 0k is the zero k 2 wm;q matrix for k = r0 ; m: From now on (a ; b; c )T is a column of Gqr with a 2 Gqr ; b; c 2 Gqm : In order to show RV = 2 we represent an arbitrary nonzero column T r (p; u 1 ; u 2 ) 2 Gq by a linear combination of at most two columns of HV0 , see Fact 1. For a nonzero column d of Gqm we denote by w (d) the only column of the matrix Wm and by (d) the only element of F 3q with d = (d)w (d): 1) Assume p 6= 0: Since the starting code V0 has R = 2 the column p can be written in the form of either (1) or (2) with f j 2 fH0 g; v = 1; 2: We denote t = u2 0 j u1 : a) Assume p = 1 f j ; f j 2 fH0 g; see (1). For j1 6= n0 and j1 = n0 we have, respectively, T 01 01 T T (p ; u 1 ; u 2 ) = (t)(0; 0; w (t)) + 1 (f j ; 1 u 1 ; j 1 u 1 ) ; t 6= 0; 1 2 F 3q (6) T w(u1 ); 0)T + 1 (f j ; 0; 101u2 )T ; (p ; u 1 ; u 2 ) = (u1 )(0;w u1 6= 0; 1 2 F 3q : (7)

ff 1 ; (0; ; 0) f 4 ; (0; 0; )T = f f 4 + ff 5 ; (0; ; )T = f f 5 ; (0; 6= ; )T = f f 5 ; ( ; 0; )T = ( + )f 4 + f f 3 ; ( ; 0; 6= )T = f 01 01 T = ( + )f 1 + f 3 ; ( ; ; 0) f 1 + ff 4 ; ( ; ; j )T = f f 2+j + ff 4 ; = f j = 0; 1; ( ; i ; 2 )T f 30i + i ff 5 ; = f i = 0; 2: By Definition 1a), K is a 2-partition. Since f 1 = 2 f 2 + ff 3 , the subset ff 1 ; f 2 g is a Q+ -subset. Hence K is a 2+ -partition and h+ (HD ) 4: Example 2: We consider q = 5; r = 3: Let U be a [6; 3]5 code ( ; 0; 0)

T

j = 1; n0

if

Consider the partition

K = ff 1 ; f 2 g; ff 3 g; ff 4 g; ff 5 g

CODES

H0 = [f 1 1 1 1 f n ]; f j 2 Gqr ; j = 1; n0 : Let m be an integer. We put q m + 1 n0 : Denote by Wm a parity-check matrix of the [wm;q ; wm;q 0 m]q 1 Hamming code with wm;q = (qm 0 1)=(q 0 1): We form an [n; n 0 r]q RV new code V with n = 2wm;q + n0 q m ; r = r0 + 2m; and parity-check matrix (5)

D with parity-check matrix HD =

6; and

Constructions considered in this section are based on the following approach to lengthening nonbinary codes with covering radius R = 2 [6]. We use an [n0 ; n0 0 r0 ]q 2 starting code V0 with a parity-check matrix

K + h+ (H ) = min K h (H ; K); h (H ) = min K h (H ; K): Example 1: We consider q = 4; r = 3: Let be a primitive element of F 4 : The construction of [6, Theorem 5.2] gives the [5; 2]4 2 E

OF

hE (HU )

n

n

q

n

n

6=

j

; 101u1 ; j 101u1 )T + 2 (f j ; 0; 201t)T ;

1 ; 2 2 F 3q : (9)

j 1 1 1 j f 01 f 01 1 1 1 f 01 j 1 1 1 j 1 2 111 j 1 1 1 j 01 1 01 2 1 1 1 01 n

j

n

q

j f f 111 f j 0 0 111 0 j 1 2 1 1 1 n

n

n

q

(5)


2) Assume p = 0: We have

T

(0; u 1 ; u2 )

=

(u1 )(0; w (u1 ); 0)T

T;

+ (u2 )(0; 0; w (u2 ))

u1 ; u2 6= 0: (10)

In (6), (7), and (10) one summand can be absent if t = 0 or ui = 0: Columns (0; w (u1 ); 0)T and (0; 0; w (u2 ))T are taken from two left submatrices of HV0 ; see (5). Finally, there is a nonzero column p 0 2 Gqr that cannot be represented in the form of (1), see Fact 1. A column (p0 ; u1 ; u2 )T cannot be written in the form of (1) as well. So, RV = 2: The parameter m is bounded only from below, see the inequality qm + 1 n0 : Hence we can obtain an infinite family of the new codes V : If

n 01 i=1

f i g = F q

we can eliminate one or two left submatrices of HV0 and reduce n: For example, assume that V0 is an E-code and use a parity-check matrix HV0 of the form of (5) without two left submatrices. Since V0 is an E-code, for p 6= 0 we always can apply the case (2) and the relations of (8) and (9). If p = 0 we cannot use the relation of (10). But for u 1 6= 0 the situation i f i g = F q always permits us to find b with u 2 = bu1 : We have

T

(0; u 1 ; u 2 )

T

(0; 0; u 2 )

T

= (f b ; u 1 ; b u 1 ) = (f n

; 0; u2 )T

0 (f b ; 0; 0)T ; 6 0; u =

0 (f n

1

; 0; 0)T ;

u2 = bu 1 (11) u1 = 0: (12)

Constructions described below develop the considered approach to improve parameters of new codes. The restriction q m + 1 n0 is connected with the condition: i 6= j if i 6= j: But if columns f i ; f j are not used together in (8) or (9) then we can put i = j : Using a 2-partition K we can assign the same value of i to all elements of each subset. Then the bound on m takes another form, e.g.,

qm + 1 h0 2 fh(H0; K); h+ (H0 ; K); hE (H0 ; K)g:

Since h0 n0 we can reduce m, obtain i f i g = F q and eliminate the left submatrices of HV0 : We define a matrix Bm ( ) where 2 F q [ f3g is called an indicator of the matrix.

2 Bm ( ) = 1 1 2

0 0 Bm (3) = 1 2

111 111

111 111

q q

;

if

2 Fq

0 (13) q where f1 ; 2 ; 1 1 1 ; q g = F q ; 1 = 0; 2 = 1; and 0 is the zero column of Gqm ; cf. (5). Construction M (i) : The index i denotes a variant of a construction. Let q 3: A starting code V0 is an [n0 ; n0 0 r0 ]q 2 code with a parity-check matrix H0 = [f 1 1 1 1 f n ]; f j 2 Gqr ; j = 1; n0 : Let K0 be a 2-partition of fH0 g: Let m be an integer. We form an (i) m [n; n 0 r ]q RV new code V with n = n + n0 q ; r = r0 + 2m; and parity-check matrix HV , cf. (5) P (f j ) HV = [D(i) S ]; S = [S1 1 1 1 Sn ]; Sj = Bm ( j ) P (f j ) = [f j 1 1 1 f j ]; j = 1; n0 (14) (i) (i) (i) where D is an r 2 n matrix, the value of n depends on the form of the matrix D(i) , the matrix P (f j ) is an r0 2 q m matrix of q m equal columns f j 2 fH0 g; j = 1; n0 , the assignment of indicators j depends on the partition K0 as follows: if columns f i ; f j belong

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to distinct subsets of K0 then the inequality i 6= j must be true, if columns f i ; f j belong to the same subset of K0 then we are free to assign the equality i = j or the inequality i 6= j : We introduce notations for Construction M (i) : Let

B=

n

i=1

f i g:

Denote by L() the union of columns of all submatrices Sj having the same indicator j = ; j = 1; n0 : Let dk be the kth column of the matrix D(i) ; k = 1; n(i) : Denote by t jc the cth column of the submatrix Bm ( j ); j = 1; n0 ; c = 1; q m : Let h jc = (f j ; tjc )T be the cth column of the submatrix Sj : By (13), (14), h jc = T T (f j ; c ; j c ) if j 6= 3; h jc = (f j ; 0; c ) if j = 3: Let + E h0 = h(H0 ; K0 ): If K0 is a 2 -partition or a 2 -partition we denote h0+ = h+ (H0 ; K0 ) or h0E = hE (H0 ; K0 ): Let h0 be one of the values of either h0 , or h0+ , or h0E : For the set fS g we define a partition KS into h0 subsets corresponding to the partition K0 : For every j , all columns of fSj g belong to the same subset of KS : Columns of sets fSi g and fSj g belong to the same subset of KS if and only if columns f i and f j belong to the same subset of K0 : The situation when the equality i = j always holds if columns f i and f j belong to the same subset of K0 is called an “h0 -assignment.” In this case, each union L() is a subset of KS and jBj = h0 where jFj is the cardinality of a set F : Now we define a partition KS of the set fS g into 2h0 subsets partitioning every subset T of the partition KS into two subsets so that the first one consists of columns h j 1 ; hj 2 of all the submatrices Sj which belong to T and the second one consists of columns fhjk : k = 3; q m g of these submatrices. By (13), (14), h j 1 = (f j ; 0; 0)T ; h j 2 = (f j ; 1; j )T if j 6= 3; hj 2 = (f j ; 0; 1)T if j = 3: Comment 1: i) In Construction M (i) we want to obtain RV = 2 where RV is the covering radius of the new code V : As before, u = (p; u1 ; u2 )T is an arbitrary nonzero column of Gqr with p 2 Gqr ; u1 ; u2 2 Gqm : In order to prove RV = 2 we show that the column u is equal to a linear combination of at most two columns of HV , see Fact 1. We represent u by a linear combination

T

(p ; u 1 ; u 2 )

y

=

z

k (f j ; tj c )T ; p=1 k=1 y + z 2; y; z 0; p ; k 2 F 3q : pdi

+

(15)

Since the starting code V0 has R = 2, a nonzero column p can be written in the form of either (1) or (2) with f j ; f j 2 fH0 g: We consider the following situation in (15); see (2)

p = 1 f j + 2 f j ; f j ; f j j 6= j ; 1 ; 2 2 F 3q ;

2 fH g; 0

y = 0;

j1 6= j2 ; z=2 (16)

where columns f j and f j belong to distinct subsets of K0 (namely, this fact forces the inequality j 6= j ): In the case (16), the combination (15) has the form of either (8), when j ; j 6= 3, or (9), when j = 3: So, in the case (16) the form of the matrix S in (14) always permits us to find needed columns tj c : Other cases depend on the matrix D(i) and the set B and are considered in the proofs of Theorems 1–5. In these proofs we represent the column (p; u 1 ; u2 )T in the form (15). From Construction M (i) and this comment it follows that for a proof of the relation RV = 2 we should consider in theorem proofs only the cases p = 1 f j 6= 0, see (1), and p = 0: ii) The approach to obtain estimates of h(HV ); hE (HV ) is as follows: we consider all cases of the representation of (15)

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for different p; u1 ; and u2 and then construct a 2-partition, or a 2+ -partition, or a 2E -partition of the column set fHV g in (14) so that columns that are used together in considered representations belong to distinct subsets. Note that for the representations of (8) and (9) we use at most one column from each submatrix Sj : iii) Let be a primitive element of F q : We define a function ( ) = 1

if = 6 1;

( ) =

In the second variant we put

q 4; u1

=

u2

2) Assume that in (15)

p

=

see (18). If

di

2 F 3q ; ( ) 2 F 3q : (17) 2 Assume q 4: Then 6= 6= 1 and each column of S is a linear combination (with coefficients from F 3q ) of exactly two

b)

j1

jp

jv

p

1

p

v

r

D

= [Am Qm ]; Am =

Wm ; Qm = m

0

f0; 3g B F [ Then the new code V is an [n; n 0 r] 2 code with n(1) = 2wm;q ;

q

0 m 0

Wm f3g:

;

q

two variants:

; 101u1 ; j 101u1 )T hj v ) = 1 ((1 0 )h j 1 + h 0 1 v = 1 u1 6= 0; 1: (19) In the first variant it holds that y = 0; z = 1; cf. (6). For the second one we put q 4; = (( 101u 1 )01 ); see (18) with p = 101u 1 ; and have in (15) y = 0; z = 2; j1 = j2 : Finally, if t = 0; u1 = 0; then u2 = 0 (p ; u 1 ; u 2 )

T

= 1 (f j

and we apply for (15) the variants (p; 0; 0)

b)

= 1 (f j

; 0; 0)T hj v ) = 1 ((1 0 )h j 2 + h 0 1 v = (20) 6= 0; 1: Assume j = 3: If u 1 6= 0 then in (15) y = z = 1; i1 wm;q ; di = (0; w(u1 ); 0)T 2 fAm g; and we use (7). If u 1 = 0; u2 6= 0; we can apply two variants, cf. T

(7), (19)

; 0; 101u2 )T hj = 1 ((1 0 )h j 1 + h 0 1 v = 1 u2 6= 0; 1:

(p ; 0; u 2 )

T

2

d

T

0 (u1 )hk1 u 1 x = 6= 0; 1; k = 0: (22) (u 1 ) T T (0; 0; u 2 ) = (u2 )(0; 0; w (u2 )) = (u 2 )hdy 0 (u 2 )hd1 y = u2 6= 0; 1; d = 3: (23) (u 2 ) Columns h kx and h k1 (respectively, hdy and h d1 ) are from distinct subsets of KS for q 3:

0

n = 2wm;q + n0 qm ; r = r0 + 2m; h(HV ) h0 : Besides, for q 4 the new code V is an E-code with h+ (HV ) h0 + 1; hE (HV ) 2h0 : Proof: The conditions B F q [ f3g and q m +1 h0 allow us to assign distinct indicators i 6= j if columns f i ; f j belong to distinct subsets of K0 : We use the h0 -assignment. 1) Assume that in (15) p = 1 f j 6= 0; see (1). For j 2 F q we denote t = u 2 0 j u 1 : a) Assume j 6= 3: If t 6= 0 then in (15) y = z = 1; i1 > wm;q ; di = (0; 0; w (t))T 2 fQm g; and we use (6). If t = 0; u 1 6= 0; we can apply for (15) the following

1

w(u1 ); 0) = (u1 )(0;w =

r

0

m

i

2

T

0

(1)

2

(0; u 1 ; 0)

p

m

1

and apply for (15) the following variants, cf. (10):

q

(1)

0:

m

1

S

p

0 we use (20).

k

K , e.g., h = (10 )h + hh ; p 2 f2; q g; 2 F 3 ; = ( 0 ); = = (18) 6 0; 1; : Theorem 1: We give Construction M : Let q 3; q + 1 h ; m 1; h 3 m

=

01u2 )01 )

(( 1

u ;u = 6 0: Then in (15) y = 2; d 2 fA g; 2 fQ g; and we use (10). Assume u = 6 0; u = 0; or u = 0; u 6= 0: Since f0; 3g B we can put = 0; = 3; for some k; d;

a) Assume

if = 1;

columns from distinct subsets of the partition

=

(u 1 )hkx

So, we have considered all cases of the representation of (15) and have shown that RV = 2: The estimate of h(HV ) follows from (6)–(10) and (19)–(23). In (19)–(23) we take the first variant with y + z = 1: We use the partition KS into h0 subsets. Each union L() is a subset of KS : By (6) and (7), columns of L(3) and columns of Qm are not used together. Hence we can inscribe the columns of Qm to the subset L(3): Similarly, we inscribe the columns of Am to some subset L(1) with 1 6= 3: The approach of Comment 1 ii) is implemented. We formed a 2-partition K1 of the set fHV g into h0 subsets. Assume q 4: We use the partition K1 and partition some subset L(2) with 2 62 f1 ; 3g into two subsets so that the first one consists of columns hj 1 and hj 2 of all the submatrices Sj which belong to L+(2): By (18), each of these two new subsets is a Q+ -subset. So, h (HV ) h0 + 1: An indicator 2 62 f1 ; 3g exists since h0 3: The estimate of hE (HV ) follows from (6)–(10), (19)–(23), and Comment 1 iii). In (19)–(23) we consider the second variant with z = 2: These relations show that the column u can be represented by a linear combination of exactly two columns of HV : Hence the new code V is an E-code. To provide the mentioned second variant we use the partition KS with 2h0 subsets. Then, similarly to the partition K1 , we inscribe the columns of Am and Qm to convenient subsets of KS : Recall that in the second variant of (19) and (21) we put

q 4:

In Theorems 2, 3, and 5 we will give conditions B = F or B = F [ f3g: They mean that we must use all elements of F or F [ f3g as indicators in (14). In this case we have “a complete set of indicators” (CSI). For columns u ; u 2 G with u = 6 0 the property CSI always permits us to find an index b with q

q

q

q

j

1

1

u2

=

b u 1 ,

2

m q

see, e.g., (11).

Theorem 2: We give Construction

M (2) :

Let

q 3; n0 q h0 ; m 1; D = Qm ; n(2) = wm;q ; B = F q : m

(2)

= 1 (f j

v)

(21)

Then the new code V is an [n; n 0 r]q 2 code with n = wm;q + n0 q m ; r = r0 + 2m: Besides, for q 4; m 2, the new code V is an E-code with h(HV ) hE (HV ) 2h0 + 2:


Proof: The relation n0 q m is a necessary condition to have CSI with B = F q : If q m > h0 we cannot use the h0 -assignment. To obtain B = F q we assign q m 0 h0 “auxiliary” indicators j so that i 6= j when columns f i ; f j belong to the same subset of the partition K0 : In this case, a subset of the partition KS can consist of more than one union L(): We call such process a “q m -assignment.” The conditions B = F q and q m h0 allow us to assign distinct indicators i 6= j if columns f i ; f j belong to distinct subsets of K0 : =B 1) Assume that in (15) p = 1 f j 6= 0, see (1). Since f3g 2 we use (6), (19), and (20). 2) Assume that in (15) p = 0:

1683

Proof: The condition n0 q m + 1 is a necessary condition to have CSI with B = F q [f3g: If q m +1 > h0+ we cannot use the h0+ assignment. We use the q m -assignment as in Theorem 2 and besides we assign the indicator j = 3 in accordance with the request of Theorem 3. 1) Assume that in (15) p = 1f j 6= 0, see (1). a) Assume j 6= 3: We can use the relations (6), (19), and (20). b) Assume j = 3: Then f j belongs to a Q+ -subset of K0 : Hence, by Definition 1, f j = ff i + ff t where ; 2 F 3q and f i ; f t belong to distinct subsets of K0 : So, we can reduce this case to the situation of (16) and use the relations (8) and (9).

a) Assume u1 6= 0: We find b with u2 = b u1 and put in (15) y = 0; z = 2; j1 = j2 = b:

u1 u ; b 1 (u 1 ) (u 1 ) 0 (u1 )(f b; 0; 0)T u2 = bu 1 ; u1 6= 0:

(0; u 1 ; u 2 )

T

(u 1 )

=

T

f b;

(24)

By (17), for q 3 columns of (24) are from distinct subsets of the partition KS , cf. (11). b) Assume u1 = 0: Then u 2 6= 0: For m 2 we treat columns of the matrix Wm as points of the projective geometry PG (m 0 1; q ), see, e.g., [6, Secs. I, IV, V], [11], and [13]. We use the same notation for a column and the corresponding point. We pass wm01;q lines from some point k : Let k di be the dth point of the ith line with k = k0i : We form a partition KW of the point set into two subsets and

fk ; k

1:

fk

i = 1; wm01;q ; d = 2; qg:

i d:

i

i = 1; wm01;q g

2) Assume that in (15) p = 0: We can use the relations (23) and (24). For estimate of hE (HV ) we use the second variant in (19), (20), and (23) with q 4 for (19). To provide it and the relation (24) we use the partition KS with 2h0+ subsets. Taking into account (6) and the case j = 3 considered in this proof, we see that columns of the union L(3) and columns of D(3) = Qm are not used together. Hence, similarly to Theorem 1, we can inscribe the columns of Qm to a subset of KS obtained from the union L(3): Remark 2: Under conditions of Theorems 1–3 (with m 2 for Theorem 2) the new code V is an E-code for q = 3 as well as for q 4: Moreover, for q = 3; m 2, the same estimates of hE (HV ) as for q 4 hold. We did not prove these facts for q = 3 to save space. Theorem 4: We give Construction M (4) : Let K0 be a 2E -partition. Let

q 3; qm + 1 h0E ; m 1; P3 (f 1 ) D(4) = 0m ; n(4) = wm;q ; Wm 1 6= 3; B F q [ f3g

Every point of a line is a linear combination (with coefficients from F 3q ) of two arbitrary points of the same line. So we can obtain two variants for (15) where in the second one points are from distinct subsets of KW : Let, e.g., w(u2 ) = k2i : Then (0; 0; u 2 )

T

= (u2 )(0; 0; w (u2 ))

T

= 1 (0; 0; k 1 )

i T

+ 2 (0; 0; k3 )

i T

;

1 ; 2 2 F 3q :

(25)

If m = 1 then jfWm gj = 1, in (25) only the first variant is possible, and V is not an E-code. To estimate hE (HV ) we use (6), (8), (9), (24), the second variant of (19), (20), and (25), and Comment 1 iii). To provide (24) and the second variant of (19) and (20) we use the partition KS with 2h0 subsets and put q 4 for (19). To provide the second variant of (25) we use the partition KW with two subsets. The inequality h(HV ) hE (HV ) follows from Definition 1. Theorem 3: We give Construction M Let

q 3;

D

(3)

= Qm ;

n0

q

n

(3)

m

=

+1

(3)

h

wm;q ;

;

2h0

+

:

T

f3g:

= (t )(f 1 ; 0; w (t))

T

0 (t)

(0; u 1 ; u 2 )

T

+

We assume that if in (14) an indicator j = 3 then the column f j belongs to a Q+ -subset of K0 : Under these conditions the new code V is an [n; n 0 r]q 2 code with n = wm;q + n0 qm ; r = r0 + 2m: Besides, for q 4 the new code V is an E-code with h(HV )

hE (HV )

(0; u 1 ; u 2 )

: Let K0 be a 2 -partition.

m 1; B = Fq [

+ 0

where P3 (f 1 ) = [f 1 1 1 1 f 1 ] is an r0 2 wm;q matrix of wm;q equal columns f 1 2 fH0 g: Then the new code V is an [n; n 0 r]q 2 E-code with n = wm;q +n0 q m ; r = r0 +2m; h(HV ) hE (HV ) h0E +1: Proof: We use the h0E -assignment with jBj = h0E : Assume p 6= 0: Since K0 is a 2E -partition we always can use (16) that leads to (8) and (9). Now assume p = 0: We have

f 1;

0u ; 0 u 1

1

1

T

; (t) (t) t = u2 0 1u 1 6= 0: (26) T T = (f 1 ; u 1 ; 1 u 1 ) 0 (f 1 ; 0; 0) ; u2 = 1u 1 ; t = 0: (27)

In (15) it holds that y = z = 1 for (26) and y = 0; z = 2; j1 = = 1; for (27). To estimate hE (HV ) we use the relations (8), (9), (26), and (27). First, we use the partition KS with h0E subsets. Then, to provide the relation (27), we partition the union L( 1 ) (that is a subset of KS ) into two subsets so that the first is the column h 11 : To provide the relations (26) we inscribe the columns of D(4) to any union L() with 6= 1 :

j2

1684


Remark 3: Constructions M (1) ; M (2) ; and M (4) of Theorems 1, 2, and 4 are Constructions A322 ; C 121 ; and AL2 of [6, Secs. III, VI, Notation 6.1] except for some details. In Theorems 1, 2, and 4 of this work the upper estimates of h(HV ); hE (HV ) are less (i.e., better) than ones in [6, Secs. IV, VI]. In [6, Sec. VI], e.g., we have hE (HV ) 3h0 + 2(wm01;q + 2) for Construction A322 and hE (HV ) 3q m + wm01;q + 2 for Construction C 121 , but in Theorems 1, 2 of this work we have hE (HV ) 2h0 and hE (HV ) 2h0 + 2: For Construction AL2 no estimate of hE (HV ) is given in [6] but one is obtained in Theorem 4 of this work. These estimates of hE (HV ) are important for constructing new codes by iterative process when newly obtained codes are starting codes for next steps, see Section III. The improved estimates helped us to obtain a number of good code families. Theorem 5: We give Construction M (5) : Let K0 be a 2E -partition. Let the matrix D(5) be absent, i.e., n(5) = 0; HV = S; and let q 3; n0 qm + 1 h0E ; m 1; and B = F q [ f3g: Then the new code V is an [n; n 0 r]q 2 E-code with n = n0 q m ;

r

= r0 + 2m; h(HV ) h (HV ) 2h0 : E

E

Proof: As before, the condition n0 q m + 1 is a necessary condition for B = F q [ f3g: If q m + 1 > h0E we cannot use the h0E -assignment. We use the qm -assignment as in Theorem 2 and additionally we assign the indicator j = 3: Similarly to Theorem 4 for p 6= 0 we can always use the relations (8) and (9). If p = 0; u1 6= 0; we use (24). If p = 0; u1 = 0; we can apply the second variant of (23). To provide (23), (24) we use the partition KS with 2h0E subsets.

at least one code V (x): We consider gaps and the values of x for concrete code families. We use a development of the notation, e.g., + 1 h2E = 2h1E ; Mi;j;k = Mi;j + mi;j;k : We fill ni;j qm gaps by other constructions. Let

q (n; R; C ) = q0r(C)

R

n i

(q 0 1)

i

i=0

be the covering density of an [n; n 0 r(C )]q R code C : For an infinite family A consisting of [n; n 0 r(An )]q R codes An we consider the value

q (R; A) = lim

inf (n; R; A ): !1 q

n

n

Denote by nq [r; R] the least known length of a q -ary linear code of codimension r, covering radius R: Let q [n; R] be the least known covering density and let rq [n; R] be the largest known codimension of a q -ary linear code of length n, covering radius R: If we obtain an [n; n 0 r]q R new code V with n < nq [r; R] then r rq [n; R]+1 and q (n; R; V ) (1=q )q [n; R]: In examples we give nq [r; 2] for comparison. Example 3: We consider q = 4; r = 2t 0 1, and use the [5; 2]4 2 code D of Example 1, see (3). Construction M (3) for V0 = D; m = 1; gives a [21; 16]4 2 E-code S with hE (HS ) 8: We take S as the initial code of a chain with n0 = 21; r0 = 5; h0E 8; m1 = 2: Construction M (5) for m = 2 forms an [n1 ; n1 0 r1 ]4 2 E-code V1 of the first level with n1 = 21 1 42 = 336; r1 = 5 + 2 1 2 =

9; h1 2 1 8 = 16: The condition n1 = 21 1 42 4 +1 16 holds for m1 1 = 2; m1 2 = 3; m1 3 = 4: So M1 = 4; 5; 6 for j = 1; 2; 3: We obtain three E-codes V1 of the second level with n1 = 21 1 4 = 5376; 21504; 86016; r1 = 5 + 2M1 = 13; 15; 17; and h2 2 1 16 = 32: + 1 32 holds for The condition n1 1 = 21 1 44 4 m1 1 = 3; 6; k = 1; 4: We obtain four E-codes V1 1 of h = (1 0 ( ))h 1 + ( )h ( ) the third level with k = 1; 4; n1 1 = 21 1 4 ; M1 1 = 4 + m = 7 ; 10 ; r = 5 + 2 M = 19 ; 21; 23; 25: 1 1 1 1 1 1 where p = k + 1; q ; ( ) = w ( ); g (p) 2 f2; kg: The matrix

Similarly, the condition n1 2 = 21 1 45 4 + 1 32 of [6, eqs. (14), (15)] and the matrix S in this work are the same. holds for m1 2 = 3; 7; k = 1; 5; and we obtain five E-codes 0 Hence K is also a partition of f g: Estimates of h(HV ); h (HV ) V1 2 of the third level with k = 1; 5; n1 2 = 21 1 4 ; in [6, Secs. IV, VI] are right but with the help of the partition K0 = 8 ; 12 ; r = 5 + 2 M = 21 ; 23, 25, 27, 29. Finally M 1 2 1 2 1 2 the estimates of [6, eqs. (20)–(22), (25)] can be explained better we have six E-codes V1 3 with m1 3 = 3; 8; k = 1; 6; n1 3 = than it has been done in [6]. Moreover, using the partition K ; M1 3 = 9; 14; r1 3 = 23; 25; 27; 29; 31; 33: So we 21 14 and approaches of Theorems 1–5 one can improve estimates of obtained a number of codes of the third level with equal parameters. [6, Sec. IV] changing terms 3h0 ; 2jBj and 3jBj by the term 2h0 : Further codimensions r 111 of the new codes again can be equal to each other but due to it gaps in the sequences of x and r = 5+2x III. FAMILIES OF CODES WITH COVERING RADIUS R = 2 are absent if x x = 4: Gaps appear for x = 1; 3; r = 7; 11: We Remark 5: With the help of Construction M (5) we design infinite obtained an infinite family A1 of [n; n 0 r]4 2 E-codes with iterative chains of new codes. A starting [n0 ; n0 0 r0 ] 2 E-code A1 : q = 4; r = 2t 0 1; n = 21 2 4 03 ; t = 3; 5; V0 , satisfying the conditions of Theorem 5, is called an initial code. and t 7; 4 (2; A1 ) 1:938: (28) The initial code has n0 q + 1 h0 for some values m : Using Construction M (5) for m = m we obtain new [n ; n 0 r ] 2 Now we fill the gaps. Construction M (1) with V0 = D; m = 2; E-codes V of the first level with n = n0 q ; r = r0 + 2m ; forms a [90; 83]4 2 E-code, Construction M (4) with V0 = S ; m = 3; h (HV ) h1 = 2h0 : Then for values of m satisfying the gives a [1365; 1354]4 2 E-code. condition n q + 1 h1 we obtain [n ; n 0 r ] 2 EWe have codes V of the second level with n = nq = n0 q ; n4 [2t 0 1; 2] = 25 13 2 4 03 0 13 r = r + 2m = r0 + 2M ; M = m + m : Similarly, we obtain codes of the (v + 1)th level from codes of the v th level, [5, eq. (4.1)]. So, n < n4 [r; 2] for the family A1 : v 2: (Note that distinct codes of the vth level can have the same parameters.) Such iterative process produces an infinite family of Example 4: We consider q = 5; r = 2t 0 1; and take the E-code [n; n 0 r] 2 E-codes V (x) with n = n0 q ; r = r0 + 2x; x ! 1: U of Example 2 as the initial code with n0 = 6; r0 = 3; h0 The values of x form an infinite sequence with some gaps in its 6; m1 = 1: We construct a code chain using Construction M (5) Remark 4: In (13) we can put f2 ; 1 1 1 ; k g = fWm g; k = wm;q+1: Let Ku0 be a partition of fSu g into three subsets fhu1 g; fhup : p = 2; kg; fhup : p = k + 1; qm g: This partition has the following property: each column of Su is equal to a linear combination (with coefficients from F q3 ) of exactly two columns from distinct subsets of Ku0 , e.g.,

E

m

;

;

;

;j

;j

E

;j

m

;

; ;k

up

p

m

u

p

ug p

p

g p

; ;k

r u

; ;k

; ;k

;

u

u

M

; ;k

r u

M

;j

;j

; ;k

; ;k

m

; ;k

E

u

S

; ;k

; ;k

; ;k

; ;k

; ;k

; ;k

M

M

; ;k

; ;k

; ;k

; ;k

i;j;k;

t

q

E

m

i

i

i

E

i

E

m

E

q

i q

i

i;j

i;j

i;j

i

i

i;j

i;j i

m

i

E

i;j

i

i;j

i

i;j

i

i;j

m

i;j q M

t

i;j

x

beginning. There is x such that for every x

E

x we can design

similarly to Remark 5 and Example 3. We obtain an E-code

V

1

with

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 45, NO. 5, JULY 1999 1 E 2 = 6 1 5 = 30; r1 = 5; h1 12: Since 30 5 + 1 12 one has m1;1 = 2; M1;1 = 3: We get a [750; 741]5 2 E-code V1;1 of the second level with h2E 24: The condition 750 5m + 1 24 holds for m1;1;k = 2; 3; 4; k = 1; 2; 3: We have M1;1;k = 5; 6; 7; and obtain three E-codes V1;1;k of the third level with

n1

1685

x

= 4: Gaps appear for x = 1; 3; r = 8; 12: If q = 4 we have gaps for r = 8; 12; 14; 20 since 42 + 1 < 4q + 4: We got a family A3 of [n; n 0 r]q 2 E-codes with

q 4; r = 2t; n = 2qt01 + qt02 + qt03 ; t = 3; 5; and t 7 if q 5; t = 3; 5; 8; 9; and t 11 if q = 4; q (2; A3 ) 2 0 2q01 + 0:5q02 0 2q03 + q04 : (31)

A: 3

= 18750; 93750; 468750; n1;1;k = 6 1 5M r1;1;k = 13; 15; 17; hE3 48:

Further we get E-codes

V

1;1;k;u

with By (31)

r1;1;1;u = 19; 21; 23; 25 r1;1;2;u = 21; 23; 25; 27; 29 r1;1;3;u = 23; 25; 1 1 1 ; 33:

4 (2; A3 ) 1:504 5 (2; A3 ) 1:6056 7 (2; A3 ) 1:7191 8 (2; A3 ) 1:7542 9 (2; A3 ) 1:7814:

For x x = 5 gaps in the sequence r = 3 + 2x are absent. Gaps exist for x = 2; 4; r = 7; 11: We obtained a family A2 of [n; n 0 r ]5 2 E-codes with parameters

r = 2t 0 1; n = 6 2 5t02 ; t = 2; 3; 5; and t 7; 5 (2; A2 ) 2:304: (29)

A : q = 5; 2

Construction M (4) with V0 = U ; m = 2; and V0 = V1 ; m = 3; gives a [156; 149]5 2 E-code and a [3781; 3770]5 2 E-code and fills the gaps. Note that for q = 5 the code family of [6, Example 6.3] cannot be obtained since a needed [4; 1]5 2 code V0 does not exist [3, Table 6.4]. Example 5: We consider q 4; r = 2t: The construction of [1, p. 104] generalized in [5, Theorem 3.1] and [6] gives the [2q + 1; 2q 0 3]q 2 code M with parity-check matrix

111 111 HM = 111 0 111 = [f 1 1 1 f 1

1

1 12

0 1 0 0

q q2

0 0 0 1

0 0 1

2 ] ; f ; 1 1 1 ; q g 1 1 2q+1 = F q ; 1 = 0: 0

111 111 111 111

0 0 1

q

=

q for (30)

ff ; 1 1 1 ; f 0 g; ff g; ff g; ff g; ff 1

for even

q

1

q +1

q

q +2

q +3

; 1 1 1 ; f 2q+1 g

q and

K =ff ; 1 1 1 ; f 0 g; ff 0 g; ff 0 g; ff 0 g; ff ; f g; ff g; ff ; 1 1 1 ; f g q

1 q

q

4

q +1

q

q +2

3

q

q +3

2

q

1

2q +1

for odd q are 2-partitions. It is easy to see that in both cases each column f i of the subset J = ff q+3 ; 1 1 1 ; f 2q+1 g is equal to a linear combination of the columns f q+2 2 = J and f j 2 J : So J is a Q+ -subset and Kq are 2+ -partitions. Let q = 5; i+1 = i; i = 1; 4: We verified by computer that

K

5

=

ff

1

nq [2t; 2] = fq (2t) + qt04 + wt02;q for

In the proof of [6, Theorem 5.1] it is shown that the partitions

K

To fill the gaps for q 4 Construction M (1) with V0 = M; m = 2; and Construction M (4) with V0 = P ; m = 3; form [n; n 0 r]q 2 Ecodes with n = fq (8)+q+2; r = 8; and n = fq (12)+q 2 +q+1; r = 12: For q = 4 Construction M (4) with V0 = P ; m = 2; gives an auxiliary [597; 587]4 2 E-code L with hE (HL ) 11: Construction M (5) with V0 = L; m = 2; forms an E-code T with n = 9552; r = 14; hE (HT ) 22: Construction M (5) with V0 = T ; m = 3; gives an E-code with n = 611328; r = 20: Due to [5, eq. (1.5)], [6, Example 6.1] we have

; f 6 g; ff 2 ; f 5 g; ff 3 ; f 4 g; ff 7 g; ff 8 ; 1 1 1 ; f 11 g

q

= 4; 5

= 2q

01 + q 02 + q 03 :

t

t

t

We design a code chain by Construction M (5) , see Remark 5 and Examples 3, 4. We obtain an E-code V1 with n1 = fq (r1 ); r1 = 10; h1E 4q + 4: Let q 5: Then q2 + 1 4q + 4; q3 + 1 8q + 8: For j = 1; 3 we get E-codes V1;j with n1;j = fq (r1;j ); r1;j = 14; 16; 18; h2E 8q + 8; and Ecodes V1;j;k with n1;j;k = fq (r1;j;k ); r1;1;k = 20; 22; 24; 26; r1;2;k = 22; 24; 26; 28; 30; r1;3;k = 24; 26; 1 1 1 ; 34: We have

nq [2t; 2] = fq (2t) + wt03;q

q 7: So, n = fq (2t) < nq [r; 2] for the family A3 :

Example 6: We consider q 7; r = 2t 0 1: For q 7 there exists always an [nq ; nq 0 3]q 2 code W with nq < q: For example, we can treat points of a small complete cap as columns of a parity-check matrix of a code W [6, pp. 2072, 2077], [12, Sec. 1.3]. Then by [10, p. 59], [11, Table 9.3], and [15, Table 1], we have nq bq=2c + 2 for q 8 and nq 6; 6; 6; 7; 8; 9; 10; 10; 10; 12; 12; 13; 14; 14; for q = 7; 8; 9; 11; 13; 16; 17; 19; 23; 25; 27; 29; 31; 32; respectively. Using the trivial partition Construction M (1) with V0 = W ; m = 1; h0 = nq ; forms an [n; n 0 5]q 2 E-code E with n = nq q + 2; h+ (HE ) nq + 1; hE (HE ) 2nq : If 2nq q + 1 we take E as the initial code. If 2nq > q + 1 Construction M (3) with V0 = E ; m2 = 1; gives Ean [n; n 0 7]q 2 initial E-code N with n = nq q + 2q + 1; h (HN ) 2nq + 2 < q2 : Similarly to Examples 3–5, Construction M (5) designs a family A4 of [n; n 0 r]q 2 E-codes with properties

A : q 7;

is a 2+ -partition. So for q 4 we have h+ (HM ) q + (3) with V0 = M; K0 = Kq ; m = 1; 1: Construction M + h0 q + 1; gives an E-code P : We take P as the initial code with n0 = fq (r0 ); r0 = 6; h0E 2q + 2; m1 = 2; where

fq (2t)

and

4

r = 2t 0 1; t = 4; 6; and t 8; nq < q; there exists an [nq ; nq 0 3]q 2 code; n = nq qt02 + 2qt03 for q + 1 2nq ; n = nq qt02 + 2qt03 + qt04 for q + 1 < 2nq ;

q (2; A4 )

1 8

(q + 4 + 6q

01 0 11q03 )

9 with n = q 02 1 + 2; 1 (2; A ) (q + 6 + 9q0 0 4q0 0 16q0 ) 8 q for even q 8 with n = + 2: 2 for odd q

q

4

q

1

2

q

3

(32)

1686


TABLE I UPPER BOUNDS ON THE LENGTH FUNCTION l(r; 2; q )

FOR

q

Results of Examples 3–6 form Table I. For (4), and (30).

= 4 ; 5; 7

r

= 3; 4

we use (3),

Remark 6: In the binary case, constructions of [9, Theorem 3.1] and their modifications [9, Sec. 3] are effective. We can treat Constructions M (1) ; M (2) as nonbinary generalizations of these constructions. Ideas of Constructions M (3) –M (5) mainly work in the binary case. But we have no improvements of binary results from [9] with the help of M (3) –M (5) : ACKNOWLEDGMENT The author wishes to thank anonymous referees for useful and ¨ detailed remarks. The author is grateful to P. R. J. Osterg˚ ard for fruitful discussions and to A. Barg for useful editorial comments. REFERENCES

By (32)

8 (2; A4 ) 1:88 9 (2; A4 ) 1:707 11 (2; A4 ) 1:943: Besides, since

n7

= 6,

we have

7 (2; A4 ) 2:09:

To fill the gaps Construction M (4) with V0 = E ; m = 2; forms an E-code C with n = nq q 3 +2q2 +q +1; r = 9; hE (HC ) 2nq + 1 < q 2 : Construction M (5) with V0 = C ; m = 2; gives an E-code with

n

nq q5 + 2q4 + q3 + q2 ; r = 13: For q = p2 16 we take as W the code of [6, Theorem 5.2] with nq = 3p 0 1: By [6, eq. (30)], it can be shown that hE (HW ) 4: Construction M (4) with V0 = W ; m = 1; forms an initial code. Construction M (5) gives a family A5 of [n; n 0 r]q 2 E-codes with A5 : q = p2 16; r = 2t 0 1 5; n = (3p 0 1)qt02 + qt03 ; =

q (2; A5 ) 4:5 0

3

p

9 0 17 + : 2q pq

(33)

For growing q = p2 codes of (33) have a covering density 4:5 but if we do not consider the q value as square we obtain a density of q=8, see (32). By [6, Example 6.3]

nq [2t 0 1; 2] = nq qt02 + 2qt03 + qt04 + wt04;q

if an [nq ; nq

0 3] 2 code with n q

q

< q exists and

n [2t 0 1; 2] = (3p 0 1)q 02 + q 03 + w 03 if q = p2 9: So, n < n [r; 2] for the families A4 and A5 : t

q

q

t

t

;q

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