NASA-CR-?OI861
New
Double-Byte
Error-Correcting Memory
Gui-Liang
Feng,
Codes
for
Systems Xinwen
July
Wu,
T. R. N. Rao"
16, 1996
Abstract Error-correcting or error-detecting to increase reliability, reduce service
codes have been costs, and maintain
used in the computer industry data integrity. The single-byte
error-correcting and double-byte error-detecting (SbEC-DbED) codes have been successfully used in computer memory subsystems. There are many methods to construct double-byte error-correcting (DbEC) codes. In the present paper we construct a class of double-byte error-correcting codes, which are more efficient than those known to be optimum, and a decoding procedure for our codes is also considered. Index
Terms:
theorem,
1
Double-byte
codes,
minimum
distance,
generalized
Bezout's
Decoding.
Introduction
Error-correcting
or
subsystems, data
which
integrity.
detecting [1-4].
error-detecting can
For
a linear
if its
are correct
equal double
to construct distance Let of parity
block
minimum
codes
used
known
that
the
codes
have
been
code
are
to increase
over
byte
greater
the
in computer reduce
single-byte field
be
error-
constructed
a linear
checks,
are
and
code
four, equal
correcting by adding
over
minimum
GF(q), distance
to or
semiconductor service
error-correcting
successfully
finite
is equal
than
errors
double-byte _> 5 was
C
to or
distance
useful
reliability,
used GF(q)
and
greater
costs, and
in computer
the
minimum
to or greater (DbEC)
than codes.
appropriate denote
d, then
where
parity by
respectively.
n,
r,
the
maintain
double-byte memory
of q elements, than
memory and
error-
subsystems q is a prime
of d [ @ ] byte errors and detecting [ 7 ] byte errors. Thus the minimum distances codes which are capable of correcting single byte errors and detecting double byte
correcting of linear errors
be
It is well
(SbEC-DbED)
power,
can
error-correcting
distances five.
code
is capable
of the
There
are
codes
many
A class
of codes
with
checks
to some
cyclic
and
A code
d the
code
over
GF(q),
length, where
which
methods minimum codes
[5].
number q =
2 i,
"Gui-Liang Feng, Xinwen Wu, T. R. N. Rao are with the Center for Advanced Computer Studies, University of Southwestern Louisiana, Lafayette, LA. 70504, USA. email:
[email protected], xw_cacs.usl.edu and
[email protected]. This work was supported in part by the National Science Foundation under Grant NCR-9505619, Louisiana Education Quality Support Fund under Grant LEQSF-(1994-96)-RD-A-36, and NASA Project NAG-W-4013,
with minimum U_(I)
distance
_> 5 was constructed
be a cyclic code
over F = GF(q),
with
where
parameters
q = 2 i, with
n = q2 a string
and
r = 7 [6]. Let
I = {1, (q_+q)12,, and
U = Uqm(I, F m-l) be the corresponding punctured code with length n = qm-1 defined on a (m - 1)-dimensional subspace F m-1 of F A class of codes over GF(2 i) with minimum TM.
distance
_> 5 was constructed
by adding
some
parity
checks
to U, these
codes
have
the
rn--1 parameters n = q,_-I , r < 2m + [-5--], m = 2,3,..-. And when q is odd, a class of codes with minimum distance _> 5 was also constructed by a similar method. The above codes
were constructed
by Dumer
ifqiseven, when n=q2, r_< 7, when n = q3 then that
no other
number
in the Theorems
double-byte
of parity
6 and
7 [5] respectively.
According
to Dumer,
r_< 7, when n=q3 then r < 9 and ifqisodd, when n =q2 r_< 8.... Dumer'scodes are known to be optimal in the sense error-correcting
checks.
codes
But unfortunately,
with
the same
the codes in Theorem
code
lengths
have
7 were defined
fewer
only over
GF(q), when q is odd. Dumer's method is very ingenious but is hard to read. It is known that in the computer systems the codes over GF(q) with q = 2i are useful. In the
present
GF(2i),
paper,
which
we will construct
have the same
a class
parameters
of double-byte
of Dumer's
error-
correcting
codes over GF(q)
codes
over
with q is odd.
And
we also study the decoding procedure of our codes. The organization of this paper is as follows. In section U, we review the generalized Bezout's theorem, which will be used to estimate the parameters of our codes. In section llI, we construct
our new double
error
correcting
codes.
In section
given. In section V, we give another construction of codes Finally, we make some concluding remarks in section VI.
2
Generalized
Let Vl, v2,
...,
Vp,
Bezout's and
u be n-tuple
IV, a decoding
with minimum
procedure
distances
is
d _> 5.
Theorem vectors.
If there
are p coefficients
ci such
that
u +
_-_-P=Icivi = O, where 0 is the zero vector, then we say that u is totally linearly dependent on vectors vl, v2, ..., vp. Sometimes, u may be linearly dependent on the vectors for only some of the components (i.e., locations). Then u is said to be partially linearly dependent on the vectors vi for 1 < i < p. The maximal possible number of those components (i.e., locations) can be used to measure the linear dependence of the vector u on the vectors vi, for 1 < i < p. The number of components, for which u is partially linearly dependent on the vectors, is called the dependent-degree of u on vi, for 1 _< i _ deg_:f2 >_ ...
i.e.,
>_ degxfp,
and let degzfl = m and degxf2 = n, where deg:_ft, indicates the maximal i such that the monomial xiy j is a term in fu. We define the x-resultant matrix of these p curves or polynomials as the following E × (m + n) matrix, where E ---- E.----1 P (_n -_ n -- degxf,) and s = degzfp:
a(o')
a_')
0
a (1)
0
0
a_ )
0
a_ 1)
a(m 1)
0 0
0
o a(o_) ap) 0
o
a(:) a?
5.
q =
2 i.
Let
7 and
3 are
[1, "'"
At_ Z2. _ -4-
X 1
, (X3k+l
as in Construction
. . .
_-
3VX3kT2/3-+-
3.3.
Then
we
GF(q).
we will
proof
of Theorem
Let
n = q3, where
1, /3, f12 is a basis
the
(Z 1 -4-X23-[-Z3/32)q2+q+l,
in Construction
section,
have
where
of the
3.4
have
the
r = 7k + 6,
parameters and
give
a proof
of Theorem
3.4,
we omit
the
q is a power
of an odd GF(q
space
T be a parity
GF(q).
10
d k 5. 3.4.
The
proof
of Theorem
details.
vector
z/32, (x + y_ + zt32) 2, (x + y_ + z/32)q_+q+l] over
cases of an
(Xl'4-X2,_t-X3_2)q_+q+l,
vector
r = 7k + 6,
matrix,
n = q3k+2,
3.3 is similar
3.3
.-3f-X3k+273k+l)
check
of codes The
2,
where
of the
3) over
dimensional q is a power
GF(q).
T be a parity
a sequence
where
matrix,
is a basis
n ----q3k+2, Construction
to higher
.-.,
"-'_-X3k+23'3k+l)
check
1, %...,73k+1
over The
k = 1,2,
, (Xl-_-X2"[-4-'"
T be
and
3.1
n = q3k+2,
prime,
3) over
check
and
GF(q).
matrix.
Then
let/3 Let
E GF(q [1,x
we have
3) -
+ Y3 + a code
Theorem 3.5
The
code
in Construction n :
Proof: a0
-I-
We al_
have -}- a2_
(x + y/3 2,
_4
z_2)
bo
-f-
_
+
(x + y/3 + z_2) 2 = x 2 + 2aoyz Hence
the
code
has
2xz+2a2gz+b2z D_ s) _ 3. easy
Let
to check
And
=
bl/_
the parameters
r = 8, -_- 2xy/3
x 2
-I-
and
b2_
and
2,
+ bo z2 + (2xy+ checks:
substitute
2alyz
1, x,y,
Obviously,
To
if A1 =
to prove
-< 3,
D{M,[z],[.],[.],[.]}
_< 3.
_< 3.
_2],[2_y+2a,yz+bl the
following A1
x
+
Bly
So we need
When
we substitute
obtain
three
the
equations
+
C1
=
x and
1, D{1,.,.,.,.}
need
= 0.
2.
2, y2+ to
prove
If A1 = 2, it is
+ b2z 2 + A4x
into
that
at
most
_< 3,
3 distinct
roots:
+ C2 = O,
2.
+ C3 = O, + B4y
+ Bsy
the
y of degree
system
to prove
second, If we can
of equations
+ C4 = O,
+ C5 = O.
(3.2),
third
and
fourth
equations,
prove
the
system
of these
then
the
proof
we three
is completed.
Now
it as follows. z = - A_ x - B1 y - C1 into
of degree
2 of it is ((1 -
1-A1/32
¢
0, we can
equations
(x + Y3 + zt 32)2,
AI_2)x
divide
+ c2y 2. Suppose
(x + cy) 2 = (x 2 + coxy three
d _> 5, we
:_
0,
+ bl z2 + A3 x + B3y
equations
to the
(x + cy) 2 = x 2 + 2cxy the
equation, + b2z2)3
that
has
+ boz 2 + A2x + B2y + 2aoyz
first
on
is equivalent
Substituting
then
_3
above
only
of equations
(x + y_ + z_2) 92+9+1 + A5x
because
the
Let
+ (y2 + 2xz + 2a2gz
z2],[y2+2_z+2_0y_+b2:2],[(_+yZ+.Z_)_2+q+_])
system
+ 2alyz
y2 + 2xz
part
into
+ Z234.
2,2xy+2algz+blz
prove
D{[x],N,[.],[.],[.]}
D{M,[z],[.],[.],[.])
2xy
the
them
-t- blz2)_
z,x2+2aoyz+boz
D_ s) = D{,h,,\2,,\z,,\4,_5}.
x 2 + 2aoyz
we prove
+ 2yz_3
that
z +
equations
d > 5.
+ (y2 + 2XZ)/32
q2+q+l.
D{[zl,[x2+2_0yz+b0 we need
2
has
(x+y/3+z_2)
if A1 = 3, we have
i.e.,
q3,
r = 8 parity 2, and
3.5
+ (1 -
it by that
(1-A_2)
2.
2 =
(1 -
A1_2)2(x
Let
c-
I_A,Zv
2c = co + cl/3 + c2/32,
+ doy 2) + (clxy
we considered
B1/J)/3y)
(x + y/3 - (A1 x + B1 y + C_ )._2) 2,
+ dly2)_
is equivalent
to the
+ (c2xy
and
+ d2y2)_
following
system
3-B_32_
+ _,yj
E GF(q3),
_2
then
c_ = do + dl/3 + d2/32, 2. So the
system
of equations
clxy + dly 2 + A_x + B_y + C_ = 0, x +coxy+doy 2+A_x+B_y+C_ =0, c2xy + d2y 2 + A_x + B_3y + C_ = O. This of the
system
of equations
coefficients
is equivalent
of x _, xy and
to (3.2)
y2 is not 1
Co
do
0
c_
dl
0
c2
d2
equal
if and to zero,
el c2
11
only i.e.,
if the
determinant
of the
matrix
of
,
In fact, if it is zero,then thereexist a nonzeroelementa a(cl,c2).
On
do -aco
the
other
= b E GF(q),
in GF(q),
i.e.,
2c = c0+c1¢3+c2/32,
c is a root
so c E GF(q2).
so (B1 -Alc)32This
hand, But
i3 + c = 0.
is a contradiction.
Construction
But
So the
3.6:
Let
T be a parity
have
GF(q).
Theorem
over 3.6
Proof:
The
we know proof
code
Suppose
x 2 -2ax
-b
GF(q3)NGF(q
that
such
that
1,/3,/32
are
= 0, whose
2) = GF(q), linearly
(dl,
d2) =
so c 2-2ac
=
coefficients thus
independent
are
c E GF(q), over
GF(q).
is completed. q = 2/.
check
n =q3,
r =8, q- XZ_
[1, x + y/3 + z/32, (x + y/3 + z32) q+l , (x +
where/3
3.6
xy/3
[::] Let
matrix,
in Construction
(x + y/3 + z/32) g+l = x 2 +
z2/32q+2.
equation
3) and
n = q3, where
Y3 + z32)q2+q+l] a code
of the
c E GF(q
E GF(q),
c 2 = d0+dl/3+d2/32,
has
is as in Construction
3.5.
Then
we
the parameters and
2 -[- xy/3
d_> 5.
q -[- y2fl_q+l
-]- yz_
q+ 2 +
X Z/32q
+
yz/32q+
l +
that _3_ = ao + al/3 + a2_ 2, rAq+l = bo + bl/) + b2/32, /3q+2
__
CO _[_ Cl/3
__
C2/32,
/32q = do + dl/3 + d232, /32q+1 = eo + el3
+ e2/32,
/3_q+2 = fo + fl/3 + f2/3 2. Substitute g0(x,
these
six
go(x,
Thus
into
equations
y, z) + gl (x, y, z)/3 + g2(x,
the
code
z/32) q2+q+l. equations
As has
= (1 + al)xy
g2(x,y,z)
= a_xy
has
r = 8 parity
in the
proof
at most
y/3q-z/32) (1-[-
Then
idea
q2+q+i , and
Ai/_2)q+l(x-_-,_
(x + cy) v+l
cq+l = ho+hl/3+h_/J
+ bly2
3 distinct
roots.
z + Alx
+ Bly
we
1, x, y, z, g0(x, 3.5,
+
C1
we need
=
y, z), gl(x, only
y, z) + A2x
+ B2y
+ C2 = O,
y,
-[-
B3y
+ C3 = O,
+ B4y
+ C4 = O,
x2 + 2, then
A3x
y, z) + A4x
in the
consider
q-
proof
of Theorem
its part ,
divide
(c q + c)xy (x+y/3)
3.5.
of degree it by
+ fi z2, + f2z 2.
y, z), g2(x,
to prove
Z_2) q2+q+l "_- AhX + Bhy
+
=
the
y, z),
following
(x + y3
+
system
of
O,
gl
Z)
(x + y/3 + z32) q+l
+ foz 2,
+ dlXZ
+ (1 + d2)xz
g0(x, (X,
have
+ doxz
+ (cl + el)yZ
+ b2y 2 + (c2 + e2)yz checks:
,, +sp__C__t_q+l l+All3_ j =
equation,
+ boy 2 + (Co + eo)yz
of Theorem
(x + y/3 the
above where
y, z) = x 2 + aoxy
gl(x,y,z)
g2(x,
We employ
the
y, z)/32,
+ C5 = O.
Substitute
2, which
is ((l+Al_2)x+
(1 + A1_32) q+l
+ c q+l.
Suppose
z = AlX+Bly+C1
that
and
let
(/3-bBl_2)y)
(x+ q+l =
c = +BP-2-P-_-_ 1+A13 _ E GF(q3).
cq + c =
go + gl/3 +g2/32,
q+l = (x+goxy+hoy2)+(glxy+hly2)/3+(g2xy+h2y2)32.
12
into
Similar
to the proof of Theorem
If it is zero, then
there
3.5, we have to prove the following 1 0
go gl
ho hi
0
g2
h2
exist a nonzero
gl
hi
g2
h2
element
a E GF(q)
So we have c q+l + ac q + ac = ho + ago = b E GF(q), c q2+q + ac q2 + ac q = b. Add the above two formulas, C q2+q
÷
such that
Example GF(q3). GF(q)
3.2: Then = GF(4)
h2)
=
a(gl, g2).
C q+l C q"
cq2 + c 3+B13
As in the proof of Theorem 3.2. Theorem 3.5 it is a contradiction.
(hi,
is not zero,
and (c q+l + ac q + ac) q = bq = b, i.e., we obtain cq2+q + c q+l + ac q2 = ac, so
--
a--
determinant
It shows c E GF(q), but c So the proof is completed.
2
1+A1_2.
As in the
proof of []
As in Example 3.1, let q = 22 = 4, and let _fl be a primitive element of GF(q 3) = GF(26) = {0, 1,/3,/_2,... 361, _A62}. Suppose a = /321, then = {0, 1, a, _2}.
We know
that
[GF(q 3) : GF(q)]
= 3, GF(q 3) is a 3-
dimensional vector space over GF(q). We can prove that for any ao,al, a2 E GF(q) = {0, 1, a, a2}, a0 + al/3 + a2/32 = 0 if and only if a0 = al = a2 = 0, i.e., 1,/3,/32 are linearly independent over GF(q). So 1,/3, f12 is a basis of GF(q 3) over GF(q). Now consider the code in Construction 3.6. (x+y/3+ z_2) q+l = (x 2 +axy+o'2g 2 +yz+axz+z 2) + (o_2y2 + yz + o_2xz + ayz + az2)_ + (xz + axy + a2y 2 + a2yz + z2)/32, (x + y/3 +/32) q_+q+l = x 3 ÷ x2y ÷ x2z ÷ c_2xy 2 ÷ xyz ÷ o_xz 2 ÷ cry 3 ÷ o_y2z ÷ yz 2 ÷ a2Z axy + aSy 2 + yz + axz + z2), (a2y s + yz + a2xz + ayz + az2), z2),x 3 + x2y + x2z + a2xy 2 + xyz
3.
Let H T = [1, x, y, z, (x s + (xz + axy + aSg 2 + a2yz +
+ axz s + ay 3 + ag_z + gz 2 + o'_z3] T be a parity
check
matrix. Then we have a code over GF(4) with n = 64, r = 8, and d _> 5. Let fl = 1, f2 = x, f3 = Y, f4 = Z, f5 = ( x2 + axy + a2y 2 ÷ yz ÷ a'xz ÷ z2), ]'6 = (a2y 2 + yz ÷ c_2xz + _'4z + az2), f7 = (xz ÷ axy ÷ a2y s + c_2yz ÷ z2), fs = x3 + x2y + x_z + a2xy 2 + xyz + v_xz 2 + o_y3 + ay2z + yz 2 + a2z 3. And Let P1 = (0, 0, 0), P2 = (0, 0, 1), P3 = (0,0, a), P4 = (0,0, a2), P5 = (1,0,0), P6 = (1,0, 1),P7 = (1,0, a), Ps = (1,0, a2), "",
=
=
P64 = ( a2, a2, a2) • Then
P1
=
0),
we have the following
P4
=
evaluated
=
table
Ps
......
f_
1 0
1 0
1 0
1 0
1 1
1 1
1 1
1 1
...... ......
1 a2
1 a_
1 a_
1 a2
1 m2
1 a_
f3
0
0
0
0
0
0
0
0
......
a
a
a_
a_
a_
a_
f4
0
1
a
a2
0
1
_
a2
......
a
a2
0
1
a
a2
f_
0
1
a2
a
1
0
1
a
......
0
a
0
a_
0
a2
f6
0
a
1
a2
0
a
0
1
......
a
a_
1
a2
0
a
f_
0
1
a2
a
0
0
1
1
......
0
a_
a
a
a2
a_
fs
0
as
a_
a2
1
1
1
a
......
a2
a
1
a_
a
_2
13
So the
parity
check
is
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
0
0
0
1
1
1
1
a
a
0 2
a 2
0 2
0 2
0 2
0 2
0 2
0 2
0 2
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
_2
02
02
G2
0
1
a
02
0
1
a
a2
0
1
02
0
1
o
_2
0
1
o
o2
0
0
0
a
0
02
0
02
1
02
0
0
o
a 2
a 2
0
02
o 2
o
1
0
1
o
02
1
a
o
1
_2
0
a
0
1
0
o2
02
0
a2
a
02
1
a .2
0
0
0
1
1
0
02
0
0
0 2
0
0 2
a 2
a .2
02
1
1
1
o
1
0
0
1
02
0 2
0
0
1
0 0 0
We can
generalize
Construction [1,
matrix
Xl"_
X2"_'_-"
x3k-1/3 GF(q
Constructions
3.7:
Let
" "2VX3k"[
3k-1
+ x3k32)q2+q+l] 3) - GF(q)
, (X
and
3.5 and
q3k
k =
sequence
of codes
over
Theorem
3.7
" "_-X3k'_'3k-1)
check
space
GF(q
Xzk"f
3k-1,
x3k/32)q2+q+1] have
-t-
in Construction
"_-
n =
"'"
of codes
3.8
The
q3k,
k
check over
codes
=
GF(q), basis
3.9
Let
/3 E GF(q of GF(q
H = [1,x+yT+,
(Xl
matrix,
vector
the
7
2
let H T be a parity Theorem
3.9
The
space
prime. " " ",
ak) -
GF(q
respectively.
Let
(X3k-2"[-
GF(q),
3k)
over
Then
/3 C GF(q),
we have
a
parameters and
where
"_- X2/3
where
3.8
+
d _ 5. q =
3.7 and
3.8
are
7,/3
have
1,3,,72, When
7 3
of an is
a
q is odd,
+w 7 3 , (x+yT+z7
code
Let
are
[1, xl
"'"
+ x27
, (Xzk-2
as in Construction
-4- --" +
-Jr- X3k-1/3
3.7.
"Jv
Then
we
and
3.4,
the parameters and
similar
n = q4, q is a power
GF(q).
check
2 i.
Xz/32)q2+q+l,
r = 7k + 1,
d _> 5.
to the
odd
basis
proofs
prime
matrix.
2
Then
in Construction n = q4,
or 2, and
of GF(q
we take
H
of Theorems
=
4) over
when
2
3
) q+l
, (x+y/3+_/3)
we have 3.9 r =
has 11,
14
a code
_.
over
2
q
+q+l
GF(q).
the parameters and
Let
7 E GF(q
GF(q),
1,/3,/32
4) is a
3,(x+ ,
(w + 0/3 + 0/32) q +q+l];
+w7
3.3
[1,x+yT+z72+w?
2
2, (x + y/3 + z/32) q +q+l, .:,
-_ E GF(q
GF(q)
2
Y7 + z3 '2 + w73)
as follows: odd
GF(q).
3) -GF(q),
3) over
2, 3,.-.,
in Construction
proofs of Theorems the details.
Construction
of the
3. 7 have
q+l,
n = q3k, The we omit
cases of an
(X 1 -_ X2/3"_-X3/32)q2"l-q+l,
where
r = 7k + 1,
"_- X3k"[3k-l)
T be a parity
a sequence
Theorem
Let
X2"{
dimensional
02
GF(q).
codes
3.8: (Xl
2,
3) over
1
q is a power
matrix,
is a basis
vector
n = q3k, Construction
to higher where
a parity
1, 7,"',73k-1
of the
3.6
2, 3,..-,
1 -t- X2"f-_-"
T be
1, 3, _2 is a basis
The
n =
o
d > 5.
q Is even,
, (w+0/3+0/3)
we 2
2
q
take
+q+l
];
Proof: Z"f 2
We prove
-_- W"t3)
yz72q+l
q+l
only ---- X 2
the
case
"q-xy7
of q is even,
+ XZ72
when
+ XW73
q is odd,
+ xY7 q +
7 q-t-I
+ b373, -Jr-c373,
7 q+3 = do + d17+
+ d3"73,
72q :
¢0 -]- C17 -[- e272 -_- e373,
72q+l
= fo -F f17
-t- f272
+ f373,
?,2q+2 = go + g17 + g272
+ g373,
73.
these
y, z, w) +
boy2+goz2
(x,
y,
Z, W)'7
= jo + J17 + J272 + J373, =
k 0 -_- k17
__
l0 .._
+2
into +
+lou3+aoxy+(co+
e3xz g3(x, the
the
g2 (x,
Z,
and
g3(x,
proofs
We need
above
equation, +
+ (Cl + fl)yz
y, z, w)
we have
g3(x,
y,
z,
we need
to prove
+ (c2 + f2)yz
+ elxz
w 3. To prove
that
the
following
system
+ Bly
+
-q- B2y
of equations C1
=
when
idea
has
/_1
at
+ (d3 + j3)yw
+
to prove ---- 1, :2, 3,
D OU
__ 3. As in
D{.\1,A2,...,X8}
__
3.
g, z, w) + A4 x + B4y + C4 = O,
g2(x,
y, z, w) + A5x
+ B5y + C5 = O,
g3(x,
y, z, w) + A6x
+ B6y + C6 = 0,
in the
+ Bsy
proof
-q- B3y
+ Cs
3 distinct
+ C3 = O,
W_3)
+ Bry
+ C7 = O,
q+l
3.5. and
Substitute consider
z = Alx its part
+ Bxy
of degree
q+l = ((l+A172+B173)x+(r+B17_+B2"_3)y) independent
roots.
= O.
of Theorem
(x + Y7 + z7 _ + linear
most
q- C2 = O,
gl(x,
+ B2y + C2 into are
+
yZ+zZ 2 )q2+ q+_],[_3] } _< 3,
y, z, w) + A3x
(x+yT+(Alx+B1y)72+(A2x+B2y)73) 1,7,72,73
+ i2xw
y, z, w), gl (x, y, z, w), g_ (x, y, z, w),
go(x,
w 3 + AsX
Because
+ kl)zw,
+ (1 + e2)xz
O,
(x + yfl + zfl2) q2+q+l + ATx
w = A2x
=
y, z, w) = x 2 + + (hi
+ (e3 + f3)yz
d _> 5, we have
to check
+ ilxw
+ (d2 + j2)yw
1, x, y, z, w, go(x,
it is easy
w -q- A2x
the
z72 + w73) q+l
go(x,
that
z + Alx
We employ
where
gl (x, y, z, w) =
+ (dl + jl)yw
D{ [9],[_] ,[g0(=,u,_,_)], [m (=,u,z,_)], [g_(=,u,z,_)], [g3(=,u,z,-,)],[(=+ i.e.,
(x + y')+
w)73,
= b3y 2 q- g3 z2 + 13 w2 + a3xy
theorems,
to prove
k373,
fo)yz+(do+jo)yw+eoxz+ioxw+(ho+ko)zw,
(x + y/3 + z/32) q2+q+l,
of the only
+
12") '2 -}- 1373.
+
W)72
+ (1 + i3)xw + (h3 + k3)zw. So the code has r = 11 parity checks: y, z, w),
-_- k272
117
above
y,
y, z, w) = b2y 2 + g_z _ + 12w2 + a2xy
(h2 -k- k2)zw,
+
that
+ h373,
73q
bly 2 + g_z 2 + l_u, 2 + (1 + al)xy g2(x,
+ h272
73q+l
12 equations gl
Suppose
= io + iiV + i272 + i373,
../3q+3
Substitute
d272
= ho + hi7
(x + y'_+
-F a3"_'3,
bo + b17 + b272
72q+3
is similar.
+ W273q+3.
7 q+2 ___c o -_- c17 -7L c272
:
proof
-_- YZ7 q+2 + YW7 q+3 + XZ72q
+ Z272q+ 2 + ZW72q+ 3 + XW73q + yW73q +1 + ZW73q+2 7 q = ao + a17 -F a272
go(x,
the
y27q+l
over
15
GF(q),
1 + A172
+ B173
+ cl and 2, which
is q+l.
7(= 0.
So we can
divide
the
have
above
equation
by (1+Alv2+B173)
(x + cy) q+l = x 2 -4-(c q + c)xy+
c q+l
=
no
(m2xy
-4- nit
-4- n272
-4- n2y2)7
q-
2 + (m3xy
cq+ly 2. Suppose
then
n373,
3. So the
+ Bly+C1
w + A2x
+ B2y
x 2 + moxy
And
if we let c-
that
c q + c =mo
(x + yc) q+l = x 2 + moxy
+ n3y2)7
z + Alx
q+l.
above
system
+
then
_+Bt_2+B_'_3
1+.4_,2+A2_3,
+ rn17 + m2"/2
hog
2 +
(rnlxy
of equations
q-
we
+ at3") nly2)7
is equivalent
'3, q-
to
=0,
+ C2 = O,
+ noy 2 + A'3x + B_y
+ C_ = O,
rnlxy
+ nly 2 + A_x
+ B'4y + C_ = O,
m2xy
+ n2y 2 + APsx + B'sy + C_ = 0,
m3xy
-4- n3y 2 -4- A_6x + B'6y + C_ = 0,
(x + y_ + z_2) q2+q+l + ATX + BTg + Cr = O, w 3 + Asx If we can
prove
of equations zero,
with
then
the
that the
proof
in the
+ Bsy
following
determinant
+ Cs = O. systems
of the
of equations,
matrix
of the
there
we prove
it as follows. 1
m0
no
0
mi
ni
0 all
i,j,
1
(nl,
n2, n3)
5.
theorem,
q is odd or even,
we have
linear
codes
u,ith the
q =
2',
with
a string
parameters: n=q And
when
,
r=2m+
+1,
m = 2. we have q-ary
constructed
3.1:
In the
Theorem
as follows.
Let
d_>5,
m=3,4,-...
codes with
n=q2,
Remark
and
r=7,
and
6[5], a class
Uqrn(I)
d_> 5.
of codes
be a cyclic
code
over over
GF(q),where F
=
GF(2i),
were
I = {1, (q_+q)}2 , and U = U_(I, F m-l) be the corresponding punctured code with length n = qrn-1 defined on a (m - 1)-dimensional subspace F m-1 of F Then the code U I over TM.
GF(2 i) with minimum U I has the parameters:
n Let Nrn(*)
=
distance
qm-1,
be a norm
_ 5 was constructed
r
