Abstrucr-In this paper, two-dimensional arrays of elements of an arbitrary finite field are examined, especially arrays having maximum- area matrices. We first ...
IEEE TRANSACTIONS
ON INFORMATION
THEORY,
VOL.
IT-18, NO. 6, NOVEMBER
775
1972
A Theory of Two-Dimensional Linear Recurring Arrays TAMIYA
NOMURA,
HIROSHI MIYAKAWA, MEMBER, IEEE, HIDEKI AKIRA FUKUDA, STUDENT MEMBER, IEEE
Abstrucr-In this paper, two-dimensional arrays of elements of an arbitrary finite field are examined, especially arrays having maximumarea matrices. W e first define two-dimensional linear recurring arrays. In order to study the characteristics of two-dimensional linear recurring arrays, we also define two-dimensional linear cyclic codes. A systematic method of constructing two-dimensional linear recurring arrays having maximum-area matrices is given using the theory of two-dimensional cyclic codes. These arrays, here called y/Garrays, may be said to be two-dimensional analogs of M-sequences. A yp-array of area N, x NY exists over GF(q) if and only if N,N, is equal to qN - 1 for some positive integer N. Many interesting characteristics of the y/?-array, such as the properties of its autocorrelation function and the properties of the characteristic arrays, are deduced and explained.
I. INTR~DLJOTI~N HE CODES that coding theory has treated in the past have been restricted almost exclusively to codes of one T dimension, i.e., codes that can be expressed on only one axis such as the time axis. This restriction seems to be partly due to the fact that currently available devices are not suitable for processing multidimensional information. Some types of codes that could be described as multidimensional arrays have been studied. Product codes, for example, are codes of this type. However, product codes cannot be called multidimensional codes in the sense of this paper since they have been introduced only in order to construct better one-dimensional codes. Many useful one-dimensional codes have been constructed to date and are being employed in various applications. This fact suggests that there must be multidimensional codes having distinctive properties that are analogous to those of one-dimensional codes. It is clear that the transmission of multidimensional information does not necessarily require a multidimensional code. However, multidimensional codes can play an important role in some applications. For example, a two-dimensional code conceived as a multidimensional analog of an ordinary M-sequence might be useful for detecting two-dimensional shifts of an array of symbols. Reed and Stewart [l], Spann [2], and Gordon [3] have made studies of the two-dimensional arrays of a so-called perfect map. The concept of the arrays of a perfect map is
IMAI,
AND
almost similar to that of arrays having maximum-area matrices, which are defined in Section II of this paper. Calabro and Wolf [9] have made studies on some twodimensional arrays having interesting properties and have covered many prior studies of these problems. These studies were quite interesting, but seemed to be somewhat lacking in generality. W e first define two-dimensional linear recurring arrays of elements of an arbitrary finite field. W e define also twodimensional linear cyclic codes and utilize them to analyze the characteristics of two-dimensional linear recurring arrays. A systematic method of constructing two-dimensional arrays having maximum-area matrices is presented using the results obtained through this analysis. These arrays are called yp-arrays. These yp-arrays have many interesting characteristics that can be regarded as two-dimensional analogs of M-sequence properties, but they also have some other interesting characteristics that are not present in M-sequences. Twodimensional linear recurring arrays and two-dimensional linear cyclic codes include the usual one-dimensional linear recurring sequences and linear cyclic codes. II. TWO-DIMENSIONAL
LINEAR RECURRING ARRAYS AND TWO-DIMENSIONAL LINEAR CYCLIC CODES
Two-Dimensional Linear Recurring Arrays An infinite matrix A, over GF(q) (q is an arbitrary power of an arbitrary prime number) [a,,, A,
aO,l
a0,2
al,1
=
+* * 1 3
aij E GF(q)
(1)
. . 1 is called here a two-dimensional array over GE’(q). If for arbitrary nonnegative integers I and J, the elements {aI+l,J+k I k = mandO I I< nor1 = nandO 5 k < m} of A, can be obtained from an n x m submatrix of A,
LaI+,,-l,,
...
aI+n-l,J+m-l J
by applying the linear relations Manuscript received July 23, 1971. T. Nomura and A. Fukuda are with the Institute of Space and Aeronautical Science, University of Tokyo, Tokyo, Japan. H. Miyakawa is with the Faculty of Engineering, University of Tokyo, Tokyo, Japan. H. Imai is with the Faculty of Engineering, Yokohama National University, Yokohama, Japan.
aI+l,J+k
=
n-l
m-l
z.
j&o
C.l,J.(“k) E GF(q);
ci,j(“k)aI+i,J+j,
k = m, 0 5 1 < n; 1 = n, 0 s k < m (3)
776
IEEE TRANSACTIONS
then A, is said to be a two-dimensional linear recurring array. The set of relations (3) is defined to be a two-dimensional linear recurring relation of degree (n,m). The array A, can be generated by this linear recurring relation, if an n x m matrix that is located at the position (0,O) is given initially. Equations (3) do not always represent a linear recurring relation if an arbitrary set {Ci,j(‘Sk)} of the elements of GF(q) is used as the coefficients of (3), because there are many ways of determining an element ai,j by (3), and these ways must not contradict each other. A necessary and sufficient condition for the set {Ci,j(z*k)} to define a linear recurring relation without contradiction has already been obtained [4]. It is obvious that all n + m relations of (3) can be obtained from at most [min (n,m) + l] relations: n + 1 relations determining {al+l,J+k 1 k = m and 0 I I < n or k = 0 and I = n} for n 5 m, or m + 1 relations determining {aI+l,J+k 1 k = m and 1 = 0 or 0 I k < m and I = n} for n 2 m. These [min (n,m) + l] relations may be conveniently used for the purpose of constructing linear recurring arrays. If there exist positive integers pX and pY satisfying ai,j
=
ai+px,j
=
(4)
ai,j+p,
for all i,j 2 0, then the period of A, is denoted by (p,.,p,,), where pX and pY are the least positive integers satisfying (4). An array A, is said to have maximum-area matrices of area N, x NY if, for some positive integers n, and nY, all n, x ny submatrices are different from each other and every nonzero n, x nY matrix over GF(q) appears in an arbitrary Wx + n, - 1) x (NY + nY - 1) matrix contained in A,. It is obvious that for an N, x NY matrix to be a maximum-area matrix a necessary condition is NXNY = “X”Y 1 4 . In this paper, an array having a period (p,,p,) and maximum-area matrices of area N, x NY is said to be an M-array if pX = N, and p,, = NY.
ON INFORMATION
N-l (A&)
=
C i=O
N-l
ai,j E GF(q) **.
aN-l,M-l
-I (5)
is called an N x M-array. The set V, XM of all these matrices is an NM-dimensional vector space over GE’(q). Every subspace C of V, XM is called a two-dimensional linear code of area N x M. The three-dimensional matrix G, which is obtained by piling up N x M matrices G,,G,; . .,G, is called a generator matrix of the linear code C, where K is the dimension of the subspace C and {Gi} is a basis of C. Hereafter, G is expressed as G = {gi,j} where gi,j are the elements of the K-dimensional vector space. This means that G is regarded as an N x M matrix over the K-dimensional vector space.
M-l C j=O
ai,jbi,j,
M-l
(6) A two-dimensional linear code C is said to be a twodimensional linear cyclic code, if and only if any cyclic shift of any codeword A = {ai,j} is also a codeword of C. The cyclic shift of A = {ai,j} is an N x M-array {ai+k,j+l}, where k and 1 are arbitrary integers and the subscripts u and v of au,” are calculated modulo N and M, respectively. It is convenient when studying the characteristics of twodimensional linear cyclic codes to introduce a ring K[x,y], which consists of all two-variable polynomials over GF(q). A residue class ring of K[x,y] with respect to an ideal (x” - 1, yM - 1) is denoted by A(xN - 1, yM - 1). There exists a unique polynomial of degree less than N with respect to x and less than M with respect to y in every residue class of A(xN - 1, yM - 1). This can be shown in a way similar to that for the one-dimensional case [4]. A residue class will be represented by such a unique polynomial hereafter. The ring A(xN - 1, yM - 1) is an NMdimensional vector space over GF(q) and is isomorphic to VN xM with respect to the correspondence a(x,y) c-) {ai,j}, where ai,j is the coefficient of the term x’yj of a(x,y). In this paper, polynomials are always regarded as the elements of A(2 - 1, yM - 1) if not otherwise indicated. Furthermore, a polynomial
Two-Dimensional Linear Cyclic Codes
aN-l,0
1972
where A = {ai,j} and B = {bi,j} are two elements of V, xM.) The generator matrix H = {hi,j} of this dual code is called a parity-check matrix of C, where the hi,j are elements of (NM - K)-dimensional vector space. It is evident that a necessary and sufficient condition for an N x M-array A = {ai,j} to be a codeword of C is
N-l
A = {ai,j} =
NOVEMBER
The dual code of C is defined as a two-dimensional linear code that is a null space of C. (The inner product is defined by
a(x,y)
An N x M matrix over GF(q)
THEORY,
=
M-l
C C ai,jXiYj i=O j=O
is also considered to represent a two-dimensional array A = {ai,j}. A two-dimensional linear code C is a two-dimensional linear cyclic code if and only if a(x, y) E C implies xky’a(x, y) E C, where k and I are arbitrary positive integers. It is easy to prove that a subspace C of the vector space A(xN - 1, YM - 1) is a two-dimensional linear cyclic code if and only if C is an ideal of the residue class ring A(xN - 1, yM - 1)
[41.
Now, an infinite two-dimensional array of a finite array A can be regarded as an array A,. It is evident that such an array A has a period. Hereafter such an array A, is also, for brevity, denoted by A or a(x,y). The array A(= A,) can always be generated by means of some linear recurring relation if A is a codeword of a two-dimensional linear cyclic code. For example, parity-check polynomials of C [cf. (6)] can be used as the linear recurring relation.
NOMURA
f?t a[. : TWO-DIMENSIONAL
LINEAR
ARRAYS
Examples: Let q = 2, n = 3, and m = 2. i) The array shown in Fig. 1 is obtained from the linear recurring relation af+3,5
= aI,, + a1t2,J
aI,Jt2
=
aI,.I
+
aI,Jtl
+
(7)
aIt2,Jt1.
This array is an M-array. ii) In Fig. 2, the array obtained from the linear recurring relation (8) is shown. aIt3,J
=
aI,J
+
aI,Jt2
=
aI,Jtl
f
aIt3,Jt1 aI,Jt2
+
Fig. 1. Example of an M-array (A is a maximum-area matrix).
aIt2,J aItl,J
+
(8)
aI+2,Jtl.
This array has maximum-area matrices but is not an Marray. iii) From the linear recurring relation (9) the array that is shown in Fig. 3 is obtained. This array has no maximumarea matrices. aIt3,J
x
=
aI,Jtl
aI+l,J
=
aI,J
+
aI,Jtl
=
aI,J
+
aI,J+l
+
+
110010010 110111011 001101010 111111000 001000011
Fig. 2. Example of an array having maximum-area matrices (A is a maximum-area matrix).
aIt2,J
aItl,Jtl
+
aIt2,J+1.
(9)
7
111. @ARRAYS
Dejinitions Let n, m, p, and J. be positive integers that satisfy the following three conditions : dlq” if4”
-
1
WV
- ’ qk - 1 thenk P
2 n
-1 A, 4 q” - 1
(12)
where q is a power of an arbitrary prime number and a/b denotes that b is divisible by a. Then, y and p are defined by y = &z”“-
l)/(q”- 1)lP
p = CPA,
(13)
where c( is a primitive element of GF(q”m) and 11is a positive integer relatively prime to q”” - 1. It is obvious that y is an element of GF(q”). It is easy to show that the orders of y and j? are given respectively by nm e
Y
eS =
-q”-1
P
4
-1 ;1
x Fig. 3. Example of an array having no maximum-area matrices.
(11)
nm
gcd
V
0011110 0011300 0110101 00l0001 0010100 1101111
(14)
Let h,(x) be a minimal polynomial of y over GF(q) and /Z&J), a minimal polynomial of p over GF(q”). Then h,(x) and As(y), whose degrees are given by Lemma 1, are unique except for multiplication by constants, i.e., multiplication by nonzero elements of GF(q) and GF(q”), respectively [5]. Lemma 1: The minimal polynomials h,(x) and h@(y) are irreducible polynomials of degrees n and m, respectively. Proof: The degree of h,(x) is given by the least positive integer d satisfying p(qd - 1) = 0 mod (q” - 1) [5]. It follows directly from (11) that d = n.
The degree of h&) is given by the least positive integer d satisfying $[(q”)d - l] = 0 mod ((9”)” - 1) [5]. This congruence is equivalent to A(qnd -
1) = 0 mod (q”” -
l),
(15)
because gcd (q, q”” - 1) = 1. Clearly A < q” since it follows immediately from (10) that 1 1 q” - 1. Therefore, A(qnd - 1) I Aqnd - 1 < q”” - 1 if we suppose that d I m - 1. Then from (15), A(q”d - 1) = 0, which is a contradiction. Consequently, d must be greater than or equal to m. Then, the degree of h&) is m since Q.E.D. d = m satisfies (15). Let P,-l = {a(x)} be a set of all polynomials over GF(q) of degrees less than n, and let a(x) E P,,- 1 correspond to a(y) E GF(q”). This correspondence between the elements of P,, _ I and the elements of GF(q”) is a one-to-one correspondence because h,(x) is an irreducible polynomial of degree n. Applying this correspondence, let the coefficients ci E GF(q”) of
correspond to ci(x) E P,- I and let m
,c-ci(x)Y i &=”
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be represented by hYp(x,y). Notice here that h,&,y) = h,(y). Then a set of all elements f(x,y) of A(xey - 1, Y ea - 1) satisfying the following simultaneous equations (16) and (17) in A(xey - 1, yes - 1) is called here a yp-array code and is represented by C,,: h,Wf(XYY) &?(&Y)f(X,Y)
= 0
(16)
= 0.
(17)
Every nonzero codeword of C,, is called a $-array.
It is evident from the definition that Cyp is a two-dimensional linear cyclic code of area eY x eS. Iff(x, y) is expressed as
THEORY,
NOVEMBER
1972
a(x,y) can be decomposed as &A
= h,&,y)al(w)
+ hy(x)a2(x,Y) + a3(xA
(20)
where the degree of a3(x, y) with respect to y is less than m. If a(y$) = 0 then a3(y,/?) = 0 from (20). Consequently, a,(y, y) must be zero. Therefore a,(x, y) can be decomposed as a3(x, y) = h,(x)a,(x, y). Thus 4w)
= hy,kw)al(x,A
+ h,(xHa2(x,A
From (16) and (17), a(x,y)f(x,y)
Structures of a y/?-Array
+ a&w>>.
= 0.
Q.E.D.
From this lemma it follows directly that the least positive integers i,j satisfying a-(X,Y)
= f(X?Y)
Y-Y&Y)
= fG%Y)
are eY and ea, respectively, wheref(x,y) is a y/?-array. This means that the period (p,,p,) of a yp-array is
then from (16) we have 0 I i < ea.
(p,,p,)
Consequently the fi(x) are codewords of a conventional one-dimensional cyclic code C, over GF(q) whose paritycheck polynomial is h,(x). Let a(x) E C,; then a(y) E GF(q”) and the correspondence a(x) e a(y) is a one-to-one correspondence. [If a(x), b(x) E C,, and a(x) = b(x) then a(y) = b(y). Conversely if a(y) = b(y) then g(x) = a(x) - b(x) has all e,th roots of unity as its roots. Therefore a(x) = b(x).] Accordingly, the correspondence f(x,y) of(y,y) is a one-to-one correspondence, where f(x, y) are the elements of A(xey - 1, Y ep - 1) that satisfy (16) and f(y, y) are the one-variable polynomials over GF(q”). We can easily show that the correspondence between the solutions f(x, y) of (16), (17) and the solutions f(y, y) of hp(~>f(r,~>
ON INFORMATION
= 0 mod (Y”” - 1)
= (“,‘)
.
(21)
The following lemma is very important for the purpose of detailed discussion of the structures of yfl-arrays. Lemma 3: Let f(x, y) be a y/?-array; then x’y’f(x,
y) = f(x, y),
0 I
i < px, 0 I j < py
(22)
if and only if i = -$k
j = 4
mod pX,
nm
q” - 1
pk,
0 I k < f$.
(23)
Proof: From Lemma 2, a necessary and sufficient condition for the validity of (22) is
(19)
is a one-to-one correspondence, taking notice of the fact that the correspondences P,- I o GF(q”) and C, c> GF(q”) mentioned previously are both isomorphisms. In particular f(x,y) = 0 corresponds tof(y, y) = 0. The congruence (19) has q”” solutions, and therefore CYsalso has q”” codewords, because the degree of h,( y) is m. Lemma 2: A necessary and sufficient condition for a polynomial a(x, y) to satisfy a(x, y)f(x, y) = 0 is a(y$) = 0, wheref(x,y) is a y/3-array. Proof: From (19) f(y,P) # 0. (If f(yJ3) = 0 then f(y,y) has all epth roots of unity as its roots. Therefore f(y,y) must be zero and then f(x, y) = 0.) Consequently, if a(x,y)f(x,y) = 0 then a(y,P) = 0. Let us regard hrp(x,y) as a polynomial in a variable y and let the coefficient of the term y” be c(x). There exists a polynomial C(x), c(x)Z(x) = 1 mod h,(x) because c(x) is a nonzero polynomial of degree less than n. Then the coefficient of the term y”’ of Z(x)hY8(x,y) takes the form 1 + k(x)h,(x). Consequently, any polynomial a(x,y) is divisible by a polynomial {i;(x)hYB(x,y) - k(x)h,(x)y”} when they are both regarded as polynomials in y. Therefore,
F)
yipj = 1.
(24)
Therefore, /I’ = ymi. Taking the e,th power of both terms, (fl’y)j = 1. Thus j must be divisible by the order of /I’?, which is denoted by ePr. Obviously epYis given by 4
e/J
nm -1
eSy = gcd (e,,e& = J. gcd [(q” - 1)/p, (4’“’ - O/4 ’ while gcd
c,
-1
P
q”“-1
~
= 4”s
2
gcd
1, 4
4
q” - 1
- 4” - l PA because gcd and pA 1 q” - 1. Therefore, eSr = (4”” -
lMqn
nm -1
-
1).
/I
NOMURA
etal.:
TWO-DIMENSIONAL
LINEAR
119
ARRAYS
Let j = ke,,. Then from (13) yipj = yipkepy = Yi+nlke Therefore (24) is valid if and only if i = -qAk mod e,, j = keSv mod eB,
(25)
k = O,l;...
Q.E.D.
Equation (23) can be derived directly from (25).
of CYs is given by (28). Finally, we can express C,, as in (27), using an arbitrary nonzero codeword f (x, y) of C,,. The structure of a y/I-array can be discussed in more detail by investigating the generator matrix of C,,. Lemma 6: Let S be a subset having nm elements of the set of all ordered pairs of two integers i and j (0 I i < pX, 0 I j < p,). Suppose the nm elements {gi,j 1 (i,j) E S} of the generator matrix G = {gi,j} of C,, are linearly independent over GF(q). Let f (k*r)(x, y) be defined as
Now, N, and NY are defined by N, = ey = px = 4”-1 P
Px-1
N, = esy = q”“p. qn - 1
f’“~‘Yx, y> =
(26)
CoroZZary 2 : Let f ‘(x, y) be the part of a y/?-array f (x, y) that contains all terms of.f(x,y) whose degrees are less than NY with respect to y. Then f (ii y) can be expressed as
PY-1
Jo
jJlo fi,j(k’z)XiYj
= xkY!f(x,Yh
(29)
where f (x, y) is an arbitrary nonzero codeword of C,,. Let wkl be the following ordered sets: w
kz = {fi,i(k*‘) I (i,j) E S}.
K-l
fc?Y)
Xz(k)yNykf'(X,
y),
where Z(k) = -r$k mod pX and K = (q” - l)/pA. From Lemma 3, x’yjf (x,y) = f (x,y) if and only if i = j = 0 when i and j are restricted to 0 I i < N, and 0 < j < NY. Furthermore N,N, = q”” - 1 and there are 4 nm codewords in C,,. Thus we have proved the following lemma. Lemma 4: C,, can be expressed using its nonzero codeword f (x, y) as
Then in the set {wkl 10 I k < N,., 0 I j < N,,} every ordered set of nm elements of GF(q), except for the set that contains no nonzero element, appears once and only once. Proof: Let us define the ordered set w, for a(x, y)(E C,,) by O, = {ai,j 1 (i,j) E S}. Then in the set {ma 1 a(x,y) E C,,} every ordered set of nm elements of GF(q) appears once and only once, since the dimension of C,, is nm and the nm elements {gij 1 (i,j) E S} of G are linearly independent over GF(q). From this result and Lemma 4, this lemma follows Q.E.D. directly.
cyfl =
The Principaz
{xiYif(x7Y)
=
,&
1 0 5 i < N,, 0 5 .i < NJ
U (0).
(27)
Now, let us derive the generator matrix of C,,. Let e,, be a set of all a”(~,y) such that
where a(x, y) is a polynomial that satisfies a(x, y)f (x, y) = 0 for a y/3-array f(x,y). It can easily be shown by methods similar to those of one-dimensional coding theory that & is the dual code of C,,. From Lemma 2, px-1
PY-1
belongs to cYs if and only if b(y-‘J-l)
Consequently, the parity-check matrix generator matrix of CYs, is expressed as G = (gi,j>
= (y-ip-j>
The equation n-l
Li(x,y) = xPxyPra(x-‘,y-‘),
= 0, i.e.,
of c,,,
= {CI-%i-q~j},
i.e., the (28)
where G is a pX x pY matrix over GF(q”“). The principal results thus obtained are summarized in the following theorem. Theorem 5: The y/?-array code C,, over GF(q), which is defined by positive integers n,m,p,A satisfying (lo), (1 l), and (12), by a positive integer q that is relatively prime to nm - 1, and by a primitive element CLof GF(q”“), is a 4 two-dimensional linear cyclic code having q”” codewords. A codeword of C,, is a pX x p,, array. A generator matrix
Theorem
C i=C)
n-l
m-l C j=O
&,jgi,j
=
C i=O
m-l C
~i,jy-'B-'
=
0,
li,j
E
GF(q)
j=O
is valid if and only if all Ai,j are 0, because the degree of a minimal polynomial of /3 - ’ over GF(q”) is m and the degree of a minimal polynomial of y-l over GF(q) is n. That is, the nm elements {gi,j 1 i = 0, 1, * * *, n - 1 and j = 0, 1,. . . , m - 1 } of G are linearly independent over GF(q). From this fact and Lemma 6, we can obtain the most important theorem in this paper by setting n, = n and n,, = m in the definition of arrays having maximum-area matrices. Theorem 7: A y/?-array has maximum-area matrices of area N, x N, = (52)
x (5
p) .
In particular, if ~Lll = q” - 1, then the y/?-array is an M-array. This array is called a y @ M-array. The following corollary is evident from the definition of the maximum-area matrix and Corollary 1. Corollary 2: In an arbitrary N, x N,, array that is included in a y/I-array f (x, y), 0 appears q”‘“-I - 1 times and any other element of GF(q) appears qnmml times. In a period of f(x,y), 0 appears (qnmml - l)(q” - 1)/p,? times and any other element of GF(q) appears qnm-l(qn - l)/pA times.
180
IEEE TRANSACTIONS
The Linear Recurring Relations Generating the y/l-Arrays The y/?-array can be generated also by a linear recurring relation, which is derived from (16) and (17). Select minimal polynomials h?(x) and hS(y) whose constant terms are 1 and y”-l, respectively. That is, h,(O) = 1 and h,(O) = y”-‘. Thus, IzYa(x,O) = x”-i. Then hi is defined as the coefficient of the term xi of h”,(x) = x”h,(x-‘) and hij is defined as the coefficient of the term xiyj of !&(x,y) = x”-lymhyS(x-l,y-l). Equations (16) and (17) can be rewritten as n-1
f k+n,l
=
f k,l+m
=
-igo
&fk+i,l
n-l
- igo
m-l
jgo
‘i,jfk+i,l+i~
where f,,, is the coefficient of the term x”y” of the yg-array f(x,y) and the subscripts u and u are calculated modulo pX and p,,, respectively. Using (30) and (31), all fi,j can be determined successively if initial values {fi,j 1 i = 0, 1,. . . , n - 1 and j = 0, l;**, m - l} are given. Example: Let q = 2, H = 3, m = 2, p = 1, 2 = q” 1 = 7, q = 1, and ~1~+ c1 + 1 = 0. The yfi-array defined by, those parameters is an M-array since ~1 = q” - 1. Further pX = N, = (q” - 1)/p = 7, p,, = N,, = (q”” - l)/ ,? = 9. The minimal polynomial of y = ~1~ over GF(2) whose constant term is 1 is by(x) = x3 + x2 + 1 and the minimal polynomial of /I = a7 over GF(23) whose constant term is yz is h,(y) = y2(y2 + (y + 1)y + 1). Thus hYB(x,y) = x2y2 + y + x2. Consequently h”,(x) = x3 + x + 1 and /&(x,y) = y2 + x2y + 1. Equations (30) and (31) are expressed as
f k+3,1
=
fk,l
+
fk+l,l
f k,Z+2
=
f k,l
+
fk+2,1+1*
From this linear recurring Fig. 4 is obtained.
relation
y/?-Arrays Having Maximum-Area K x (qN - 1)/K
ON INFORMATION
110010103 111010101 111100011 001000001 000110110 110100010 001110111
Y
THEORY.
NOVEMBER
1972
DFig. 4.
Example of a yg-M-array (q = 2, n = 3, m = 2, p = 1, 1=7,q=l,a6+a+l=O).
(qN - 1)/K) = 1. At least one such A exists. For example, let 1 be equal to 1. Then it can be shown easily that these n, m, ~1,and 2 satisfy (IO), (1 l), and (12). Let q be an arbitrary positive integer relatively prime to qN - 1 and let c1 be an arbitrary primitive element of GF(qN); then the $-array constructed using these parameters has maximum-area matrices of area K x L. If K is relatively prime to L, then A can be equated to K. Then, FA = q” - 1, and in this case the y/?-array previously described is a yP M-array. A Correspondence Between y/3-Arrays and M-Sequences The minimal polynomial of CLover GF(q) is denoted by The cyclic code C, of length q”” - 1 that is defined by the parity-check polynomial h,(z) is called the Msequence code [S] or the shortened first-order Reed-Muller code [6]. The following theorem indicates a correspondence between the cyclic code C, and the y/I-array code C,,. Theorem 9: Let Pe,- 1 be the set of all polynomials a(z) over GF(q) whose degrees are less than e,, where e, = nm 4 - 1, and let P be the set of all polynomials a’(x, y) over GF(q) whose degrees are less than N, with respect to x and less than N,, with respect to y. Furthermore, let Q be the set of all polynomials a(x,y) defined by the following equation : h,(z).
K-l
the y/I M-array
@,y) of
=
x’(k)yNyka’(x, y),
C k=O
l(k) = -qAk Matrices of Area
If a yfi-array over GF(q) is constructed, there must be a positive integer N that satisfies the relation N,N, = qN - 1. Furthermore, if the y/?-array is a y/I M-array, N, must be relatively prime to NY, since N, is equal to ;1 in that case. Now, as a converse of the results so far obtained the next theorem can be derived. Theorem 8: A y/?-array having maximum-area matrices of area K x L exists if K is an arbitrary divisor of qN - I and L = (qN - 1)/K for some positive integer N. Furthermore a y/l M-array having maximum-area matrices of area K x L exists if in addition to the previously mentioned conditions K is relatively prime to L. Proof: Let n be the least positive integer satisfying the conditions K 1 q” - I and n 1 N. It is obvious that such an integer n always exists. Then let p be defined by p = (q” - 1)/K and let k be a divisor of K where gcd (A,
mod pX, IC = (4” - l)/pJ.
Now, let 8 be a mapping from Q in P,=- 1 defined by 8: a(x,y)
+ a(z) = a’(zNY,z4’) mod (z+ - l),
where a’(x, y) E P is the part of a(x, y) whose degree with respect to y is less than N,,. Then 8 is a one-to-one mapping from Q onto P,.- 1. The y/3-array code C,, is contained in Q and the image of CYs is an M-sequence code C,. Conversely, the inverse image of any M-sequence code of length e, that is included in P,.- I is a @array code. Proof: It is obvious that 8 is a composition mapping B, 0 B,, where
e1 : 4x,
Y> + a’(x,y>
8,: a’(x,y)
+ a(z) = a’(zNY,zqrl) mod (zea - 1).
Obviously 8, is a one-to-one mapping onto P. On the other
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hand, 8, transfers the coefficient of the term x’y’ (0 < i < N,, 0 I j < NJ of a’(x,y) to the coefficient of the term z’ (I = NJ + $j mod e,) of a(z). Now let I, = NJ1 + $jj, mod e,, IfI,
0 5 i,, i2 < N,; =
1. Meanwhile, from the Proof of Lemma 3 (25) this is valid if and only if il = i, and j, = j, when i,, jl, i,, and j, are restricted by -N, < i, - i, < N,, -NY < j, - j, < NY. Furthermore N,N, = q”” - 1 = e,. Consequently e2 is a one-to-one map from (i,j) (0 I i < N,, 0 I j < NY) to 1 (0 I I < e,). This means that the coefficients of a(z) are obtained by the rearrangement of the coefficients of a’(x,y). Thus it is shown that 0 = e2 o 8, is a one-to one mapping from Q onto P,=- 1. From Corollary 1 the codeword f (x, y) of C,, is contained in Q. The element of the generator matrix G of C,, that corresponds to the coefficient fi,j of f (x, y) is a-Nyi-tlLj = 6’ and the image of C,, has the following generator matrix : aocl-1a-2..
. u-e,+l
,y =
igo
Pr-1 j50
4(ai,j~ad
From (33), s is always positive. a E Wq) by
I2 = Nyi2 + $j2 mod e,, 0 I jl, j, -c N,. = I,, uZ~ = u12sThus yilpjl = yi2pj2, i .e., yil-i2pjl-j2
[
tively. Here s is given by px-1
x(a)= J&
Let us define x(a) for
44h4
a E GE’(q).
(36)
Further let “m-ix(l)
a=q
- 4(0,0).
(37)
Then, the autocorrelation functions of y/&arrays can be given by the following theorem. Theorem 10: The autocorrelation function pg(z1,z2) of a y/3-array in a period (i.e., for 0 I r1 < p,., 0 I z2 < p,) is given as follows. If 21
=
-qlk
+ k-,el mod pX,
z2 = N,,k + tc,el mod p,,,
0 I
12
k < (q” - 1)//A o 0
(33) (34)
The actual form of the function 4 will be determined by the physical meaning of the elements of GF(q). In most cases, however, the previously mentioned restrictions will be adequate. The autocorrelation function p+(z, ,z,) of an array a(x, y) is defined by P4(71372)
=
px--l
Py-1
igo
j&
~(ai,j,ai+rl,j+r2)lS,
(35)
where (p,,p,) is the period of the yfl-array and the subscripts u,v of au.” are calculated modulo pX and p,,, respec-
(41)
of (35) is rewritten as
where 4 is assumed to satisfy the three conditions: 4(G)
- 1)/d.
p&,,zz)
PY-1
= igo jgo 4Is,
where f(‘,j)(x, y) was defined previously in (29) and fk,l(i’j) is the coefficient of the term x”y’ of ,f’i3j)(x,y). From Corollary 1, (37), and (41) we have N,-1
N,-1
(42) Suppose y”fi” Thus
E GF(q); then from Lemma 2
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From Corollary 2 and (36), it can easily be shown that pg(zl ,T~) becomes = (4 “m- ‘x(y”P’“)
P&J,)
- ~(O,O)}/o.
then = el mod (4”” - 1).
both sides by
K~
Multiplying 71 -
~/ZK,Z,
+
(43)
and using (39) we have
~~qA.t.2
=
rc,el
mod (4”” -
1).
From this equation Tl
because 1, 1 q”“’ -
THEORY,
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not difficult to prove these properties, if one recalls the method of constructing $-arrays. These properties are closely analogous to those of one-dimensional M-sequences
c71.
Now, let us obtain the r1 and z2 that satisfy y”fir2 E GF(q). If y”p” = 61 = g’, 012
