On various parameters of Zq-simplex codes for an even integer q
In this paper, we defined the Zq-linear codes and discussed its various parameters. We constructed Zq-Simplex code and Zq-MacDonald code and found its parameters. We have given a lower and an upper bounds of its covering radius for q is an even integer.
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irk-123456789-1542572019-06-16T01:27:01Z On various parameters of Zq-simplex codes for an even integer q Chella Pandian, P. Durairajan, C. In this paper, we defined the Zq-linear codes and discussed its various parameters. We constructed Zq-Simplex code and Zq-MacDonald code and found its parameters. We have given a lower and an upper bounds of its covering radius for q is an even integer. 2015 Article On various parameters of Zq-simplex codes for an even integer q / P. Chella Pandian, C. Durairajan // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 242-253 . — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC:94B05, 11H31. http://dspace.nbuv.gov.ua/handle/123456789/154257 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper, we defined the Zq-linear codes and discussed its various parameters. We constructed Zq-Simplex code and
Zq-MacDonald code and found its parameters. We have given a lower and an upper bounds of its covering radius for q is an even integer. |
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Chella Pandian, P. Durairajan, C. On various parameters of Zq-simplex codes for an even integer q Algebra and Discrete Mathematics |
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Chella Pandian, P. Durairajan, C. |
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Chella Pandian, P. |
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On various parameters of Zq-simplex codes for an even integer q |
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On various parameters of Zq-simplex codes for an even integer q |
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On various parameters of Zq-simplex codes for an even integer q |
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On various parameters of Zq-simplex codes for an even integer q |
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On various parameters of Zq-simplex codes for an even integer q |
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on various parameters of zq-simplex codes for an even integer q |
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Інститут прикладної математики і механіки НАН України |
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On various parameters of Zq-simplex codes for an even integer q / P. Chella Pandian, C. Durairajan // Algebra and Discrete Mathematics. — 2015. — Vol. 19, № 2. — С. 242-253 . — Бібліогр.: 18 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT chellapandianp onvariousparametersofzqsimplexcodesforanevenintegerq AT durairajanc onvariousparametersofzqsimplexcodesforanevenintegerq |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 19 (2015). Number 2, pp. 243–253
© Journal “Algebra and Discrete Mathematics”
On various parameters of Zq-simplex codes
for an even integer q
P. Chella Pandian∗ and C. Durairajan∗∗
Communicated by V. Artamonov
Abstract. In this paper, we defined the Zq-linear codes and
discussed its various parameters. We constructed Zq-Simplex code
and Zq-MacDonald code and found its parameters. We have given
a lower and an upper bounds of its covering radius for q is an even
integer.
1. Introduction
A code C is a subset of Zn
q , where Zq is the set of integer modulo q
and n is any positive integer. Let x, y ∈ Z
n
q , then the distance between x
and y is the number of coordinates in which they differ. It is denoted by
d(x, y). Clearly d(x, y) = wt(x − y), the number of non-zero coordinates
in x − y. wt(x) is called weight of x. The minimum distance d of C is
defined by
d = min{d(x, y) | x, y ∈ C and x 6= y}.
The minimum weight of C is min{wt(c) | c ∈ C and c 6= 0}. A code
of length n cardinality M with minimum distance d over Zq is called
(n, M, d)q-ary code. For basic results on coding theory, we refer [16].
∗The first author would like to gratefully acknowledge the UGC-RGNF[Rajiv Gandhi
National Fellowship], New Delhi for providing fellowship.
∗∗The second author was supported by a grant(SR/S4/MS:588/09) for the Depart-
ment of Science and Technology, New Delhi.
2010 MSC: 94B05, 11H31.
Key words and phrases: codes over finite rings, Zq-linear code, Zq-simplex code,
Zq-MacDonald code, covering radius.
244 Zq -s implex codes for an even integer q
We know that Zq is a group under addition modulo q. Then Z
n
q is a
group under coordinatewise addition modulo q. A subset C of Zn
q is said
to be a q-ary code. If C is a subgroup of Zn
q , then C is called a Zq-linear
code. Some authors are called this code as modular code because Z
n
q is
a module over the ring Zq. In fact, it is a free Zq-module. Since Z
n
q is a
free Zq-module, it has a basis. Therefore, every Zq-linear code has a basis.
Since Zq is finite, it is finite dimension.
Every k dimension Zq-linear code with length n and minimum distance
d is called [n, k, d] Zq-linear code. A matrix whose rows are a basis elements
of the Zq-linear code is called a generator matrix of C. There are many
researchers doing research on code over finite rings [4, 9–11, 13, 14, 18].
In the last decade, there are many researchers doing research on codes
over Z4 [1–3,8, 15].
In this correspondence, we concentrate on code over Zq where q is
even. We constructed some new codes and obtained its various parameters
and its covering radius. In particular, we defined Zq-Simplex code, Zq-
MacDonald code and studied its various parameters. Section 2 contains
basic results for the Zq-linear codes and we constructed some Zq-linear
code and given its parameters. Zq-Simplex code is given in section 3 and
finally, section 4 we determined the covering radius of these codes and
Zq-MacDonald code.
2. Zq-linear code
Let C be a Zq-linear code. If x, y ∈ C, then x − y ∈ C. Let us consider
the minimum distance of C is d = min{d(x, y) | x, y ∈ C and x 6= y}.
Then
d = min{wt(x − y) | x, y ∈ C and x 6= y}.
Since C is Zq-linear code and x, y ∈ C, x − y ∈ C. Since x 6= y,
min{wt(x − y) | x, y ∈ C and x 6= y} = min{wt(c) | c ∈ C and c 6= 0}.
Thus, we have
Lemma 1. In a Zq-linear code, the minimum distance is the same as
the minimum weight.
Let q be an even integer and let x, y ∈ Z
n
q such that xi, yi ∈ {0, q
2},
then xi ± yi ∈ {0, q
2}.
Lemma 2. Let q be an integer even. If x, y ∈ Z
n
q such that xi, yi ∈ {0, q
2},
then the coordinates of x ± y are either 0 or q
2 .
P. Chella Pandian, C. Durairajan 245
Now, we construct a new code and discuss its parameters. Let C be
an [n, k, d] Zq-linear code. Define
D = {(c0c · · · c)+α(0112 · · · q − 1) | α ∈ Zq, c ∈ C and i = ii · · · i ∈ Z
n
q }.
Then, D = {c0c · · · c, c0c · · · c+0112 · · · q − 1, c0c · · · c+2(0112 · · · q − 1)
, · · · , c0c · · · c + (q − 1)(0112 · · · q − 1) | c ∈ C and i ∈ Z
n
q
}. Since any Zq-
linear combination of D is again an element in D, therefore the minimum
distance of D is d(D) = min{wt(c0c · · · c), wt(c0c · · · c + 0112 · · · q − 1),
wt(c0c · · · c+2(0112 · · · q − 1)), · · · , wt(c0c · · · c+(q−1)(0112 · · · q − 1))
| c ∈ C and i ∈ Z
n
q
}.
Clearly min{wt(c0c · · · c) | c ∈ C&c 6= 0} > qd.
Let c ∈ C. Let us take c has ri i′s where i = 0, 1, 2, · · · , q − 1. Then
for 1 6 i 6 q − 1,
wt(c + i) =
q−1
∑
j=0
rj − rq−i.
That is wt(c + i) = n − rq−i. Therefore
wt(c0c · · · c + 0112 · · · q − 1)
= wt(c + 0) + 1 + wt(c + 1) + wt(c + 2) + · · · + wt(c + q − 1)
= n − r0 + 1 + n − rq−1 + n − rq−2 + · · · + n − r1
= (q − 1)n + 1.
Similarly, for every integer i which is relatively prime to q
wt((c0c · · · c) + i(0112 · · · q − 1)) = (q − 1)n + 1.
For other i’s
min
i∈Zq
{wt(c0c · · · c + i(0112 · · · q − 1))}
= wt(c + 0) + 1 + wt(c · · · c +
q
2
(12 · · · q − 1))
= wt(c + 0) + 1 + wt(c · · · c + (
q
2
0
q
2
0 · · ·
q
2
0
q
2
))
=
q
2
wt(c + 0) + 1 +
q
2
wt(c +
q
2
)
=
q
2
(n − r0) + 1 +
q
2
(n − r q
2
)
=
q
2
n + 1 +
q
2
(n − r0 − r q
2
).
Hence, d(D) = min{qd, (q − 1)n + 1, q
2n + 1 + q
2(n − r0 − r q
2
)}. Thus, we
have
246 Zq -s implex codes for an even integer q
Theorem 1. Let C be an [n, k, d] Zq-linear code, then the
D = {c0c · · · c + α(0112 · · · q − 1) | α ∈ Zq, c ∈ C and i = ii · · · i ∈ Z
n
q }
is a [qn + 1, k + 1, d(D)] Zq-linear code.
If there is a codeword c ∈ C such that it has only 0 and q
2 as coordinates,
then
wt(c0c · · · c + 0
q
2
q
2
0
q
2
· · · 0
q
2
)
= wt(c + 0) + 1 + wt(c +
q
2
) + wt(c + 0) + · · · + w(c +
q
2
)
= 1 + r q
2
+ r0 + r q
2
+ · · · + r0
=
q
2
(r0 + r q
2
) + 1 =
q
2
n + 1.
Hence, d(D) = min{qd, q
2n + 1}. Thus, we have
Corollary 1. If there is a codeword c ∈ C such that ci = 0 or q
2 and if
n 6 2d − 1, then d(D) = q
2n + 1.
3. Zq-simplex codes
Let G be a matrix over Zq whose columns are one non-zero element
from each 1-dimensional submodule of Z2
q . Then this matrix is equivalent
to
G2 =
[
0 1 1 2 · · · q − 1
1 0 1 1 · · · 1
]
.
Clearly G2 generates [q + 1, 2, q
2 + 1] code. Inductively, we define
Gk+1 =
00 · · · 0 1 11 · · · 1 22 · · · 2 · · · q − 1q − 1 · · · q − 1
0
Gk
... Gk Gk · · · Gk
0
for k > 2. Clearly this Gk+1 matrix generates [nk+1 = qk+1
−1
q−1 , k + 1, d]
code. We call this code as Zq-Simplex code. This type of k-dimensional
code is denoted by Sk(q). For simplicity, we denote it by Sk.
Theorem 2. Sk(q) is an [nk = qk
−1
q−1 , k, q
2nk−1 + 1] Zq-linear code.
P. Chella Pandian, C. Durairajan 247
Proof. We prove this theorem by induction on k. For k = 2, from the
generator matrix G2, it is clear that d = q
2 +1 and the theorem is true. Since
there is a codeword c = 0 q
2
q
20 q
2 · · · 0 q
20 q
2 ∈ S2 and n = q+1 6 2( q
2 +1)−1 =
2d − 1, by Corollary 1 implies d(S3) = q
2n2 + 1 and hence the S3 is
[n3 = q3
−1
q−1 , 3, q
2n2 +1] code. Since c0c · · · c+ q
2(0112 · · · q − 1) ∈ S3 whose
coordinates are either 0 or q
2 and satisfies the conditions of the Corollary 1,
therefore d(S4) = q
2n3 + 1 and hence the S4 is [n4 = q4
−1
q−1 , 4, q
2n3 + 1] code.
By induction we can assume that this theorem is true for all less than k.
That is, there is a code c ∈ Sk−1 whose coordinates are either 0 or q
2 and
nk−1 6 2dk−1 − 1. By Corollary 1, dk = q
2nk−1 + 1. Therefore Sk(q) is an
[ qk
−1
q−1 , k, q
2nk−1 + 1] Zq-linear code. Thus we proved.
Now, we are going to see the minimum distance of the dual code
of this Zq-Simplex code. Since the matrix Gk(q) has no zero columns,
therefore, the minimum distance of its dual is greater than or equal to 2.
Since in the first block of the matrix Gk, there are two columns whose
transpose matrices are (0, 0, · · · , 0, 1, 1) and (0, 0, · · · , 0, a, 1). Since addi-
tion and multiplications are modulo q and q is even, q
2(0, 0, · · · , 0, 1, 1) +
q
2(0, 0, · · · , 0, q − 1, 1) = 0. That is, there are two linearly dependent
columns. Therefore, the minimum distance of the dual code is less than or
equal to 2. Hence the dual of Sk is [nk = qk
−1
q−1 , nk − k, 2] Zq-linear code.
4. Covering radius
The covering radius of a code C over Zq with respect to the Hamming
distance d is given by
R(C) = max
u∈Zn
q
{
min
c∈C
{d(u, c)}
}
.
It is easy to see that R(C) is the least positive integer r such that
Z
n
q = ∪c∈CSr(c)
where
Sr(u) =
{
v ∈ Z
n
q } | d(u, v) 6 r
}
for any u ∈ Z
n
q .
Proposition 1 ([5]). If appending( puncturing) r number of columns in
a code C, then the covering radius of C is increased( decreased ) by r.
248 Zq -s implex codes for an even integer q
Proposition 2 ([17]). If C0 and C1 are codes over Z
n
q generated by
matrices G0 and G1 respectively and if C is the code generated by
G =
(
0 G1
G0 A
)
,
then r(C) 6 r(C0) + r(C1) and the covering radius of C satisfy the
following
r(C) > r(C0) + r(C1).
Since the covering radius of C generated by
G =
(
0 G1
G0 A
)
,
is greater than or equal to r(C0) + r(C ′) where C0 and C ′ are codes
generated by
[
0
G0
]
=
[
G0
]
and
[
G1
A
]
, respectively, this implies
r(C) > r(C0) + r(C1) because C1 is a subcode of the code C ′.
A q-ary repetition code C over a finite field Fq with q elements is
an [n, 1, n] linear code. The covering radius of C is
⌊
n(q−1)
q
⌋
[12]. For
basic results on covering radius, we refer to [5], [6]. Now, we consider the
repetition code over Zq. There are two types of repetition codes.
Type I. Unit repetition code generated by Gu = [
n
︷ ︸︸ ︷
uu . . . u] where u is an unit
element of Zq. This matrix generates Cu is [n, 1, n] Zq-linear code.
That is, Cu is (n, q, n) q-ary repetition code. We call this as unit
repetition code.
Type II. Zero divisor repetition code is generated by the matrix Gv =[
n
︷ ︸︸ ︷
vv . . . v]
where v is a zero divisor in Zq. That is, v is not a relatively prime
to q. This is an (n, q
v
, n) code over Zq. This code is denoted by Cv.
This code is called zero divisor repetition code.
With respect to the Hamming distance the covering radius of
Cu is
⌊
n(q−1)
q
⌋
[12] but clearly the covering radius of Cv is n because
code symbols appear in this code are zero divisors only. Thus, we have
Theorem 3. R(Cv) = n and R(Cu) =
⌊
(q−1)n
q
⌋
.
Let φ(q) = #{i | 1 6 i < q & (i, q) = 1} be the Euler φ-function. Let
U = {i ∈ Z | 1 6 i < q & (i, q) = 1} be the set of all units in Zq and let
P. Chella Pandian, C. Durairajan 249
O = Zq \ U be the set which contains all zero divisors and 0. Let C be a
Zq-linear code generated by the matrix
[
n
︷ ︸︸ ︷
11 . . . 1
n
︷ ︸︸ ︷
22 . . . 2 · · ·
n
︷ ︸︸ ︷
q − 1q − 1 . . . q − 1],
then this code is equivalent to a code whose generator matrix is
[u1u1· · ·u1u2u2· · ·u2· · ·uφ(q)uφ(q)· · ·uφ(q)o1o1· · ·o1o2o2· · ·o2· · ·oror· · ·or]
where r =q−1−φ(q). Let A be a code equivalent to the unit repetition code
of length φ(q)n generated by [u1u1· · ·u1u2u2· · ·u2· · ·uφ(q)uφ(q)· · ·uφ(q)],
then by the above theorem, R(A) =
⌊
(q−1)φ(q)n
q
⌋
. Let B be a code equiva-
lent to the zero divisor repetition code of length (q−1−φ(q))n generated by
[o1o1 · · · o1o2o2 · · · o2 · · · oror · · · or], then by the above theorem, R(B) =
(q − 1 − φ(q))n. By Proposition 2, R(C) >
⌊
(q−1)φ(q)n
q
⌋
+ (q − 1 − φ(q))n.
Without loss of generality we can assume that the generator matrix of
A as [111 · · · 1]. Since R(A) =
⌊
(q−1)φ(q)n
q
⌋
and C is obtained by appending
some (q −1−φ(q))n columns to A, by Proposition 1 the covering radius of
C is increased by at most (q−1−φ(q))n. Therefore, R(C) 6
⌊
(q−1)φ(q)n
q
⌋
+
(q − 1 − φ(q))n. Thus, we have
Theorem 4. Let C be a Zq-linear code generated by the matrix
[
n
︷ ︸︸ ︷
11 . . . 1
n
︷ ︸︸ ︷
22 . . . 2 · · ·
n
︷ ︸︸ ︷
q − 1q − 1 . . . q − 1].
Then C is a [(q − 1)n, 1, q
2n] Zq-linear code with R(C) =
⌊
(q−1)φ(q)n
q
⌋
+
(q − 1 − φ(q))n.
Now, we see the covering radius of Zq-Simplex code. The covering
radius of Simplex codes and MacDonald codes over finite field and finite
rings were discussed in [12], [14].
Theorem 5. For k > 2,
R(Sk+1) 6
(k − 1)(q − 1)φ(q) + (q2 − q − φ(q))(qk+1 − q2)
q(q − 1)2
+ R(S2).
Proof. For k > 2, Sk+1 is [nk+1 = qk+1
−1
q−1 , k + 1, q
2nk + 1] Zq-linear code.
By Proposition 2 and Theorem 4 give
R(Sk+1) 6 (1 +
⌊
(q − 1)φ(q)nk
q
⌋
+ (q − 1 − φ(q))nk) + R(Sk)
250 Zq -s implex codes for an even integer q
6 (1 +
(q − 1)φ(q)nk
q
+ (q − 1 − φ(q))nk) + R(Sk)
6
(
1 +
q2 − q − φ(q)
q
nk
)
+ R(Sk).
This implies
R(Sk) 6 (1 +
q2 − q − φ(q)
q
nk−1) + R(Sk−1).
Combining these two, we get
R(Sk+1) 6 (1 +
q2 − q − φ(q)
q
nk) + (1 +
q2 − q − φ(q)
q
nk−1) + R(Sk−1)
Similarly, if we continue, we get
R(Sk+1) 6 (1 +
q2 − q − φ(q)
q
nk) + (1 +
q2 − q − φ(q)
q
nk−1) + · · ·
+ (1 +
q2 − q − φ(q)
q
n2) + R(S2).
Since nk = qk
−1
q−1 , for k > 2, therefore
R(Sk+1) 6 (k−1)+
q2−q−φ(q)
q
(
qk−1
q−1
+
qk−1−1
q−1
+· · ·+
q2−1
q−1
)
+R(S2)
6 (k−1)+
q2−q−φ(q)
q
(
qk+qk−1+· · ·+q2−(k−1)
q−1
)
+R(S2)
6
(k−1)φ(q)+(q2−q−φ(q))((qk+1−1)/(q−1)−(q+1))
q(q−1)
+R(S2)
6
(k−1)(q − 1)φ(q) + (q2 − q − φ(q))(qk+1 − q2)
q(q − 1)2
+ R(S2).
Hence the proof is complete.
In particular, for q = 4, R(Sk+1) 6 5.4k+1+3k−29
18 for k > 2 because of
simple calculation R(S2) = 3.
Now, we can define a new code which is similar to the Zq-MacDonald
code. Let
Gk,u =
(
Gk \
(
0
Gu
))
P. Chella Pandian, C. Durairajan 251
for 2 6 u 6 k − 1 where 0 is a (k − u) × qu
−1
q−1 zero matrix and
(
A \ B
)
is a matrix obtained from the matrix A by removing the matrix B. The
code generated by Gk,u is called Zq-MacDonald code. It is denoted by
Mk,u. The Quaternary MacDonald codes were discussed in [7].
Theorem 6. For 2 6 u 6 r 6 k,
R(Mk+1,u)6
(k−r+1)(q−1)φ(q)+(q2−q−φ(q))qr(qk−r+1−1)
q(q−1)2
+R(Mr,u).
Proof. By using, Proposition 2, we get
R(Mk+1,u) 6 (1 +
⌊
(q − 1)φ(q)nk
q
⌋
+ (q − 1 − φ(q))nk) + R(Mk,u)
6 (1 +
(q − 1)φ(q)nk
q
+ (q − 1 − φ(q))nk) + R(Mk,u)
6
(
1 +
q2 − q − φ(q)
q
nk
)
+ R(Mk,u).
This implies R(Mk,u) 6 (1 + q2
−q−φ(q)
q
nk−1) + R(Mk−1,u). Combining
these two, we get
R(Mk+1,u) 6 (1+
q2−q−φ(q)
q
nk)+(1+
q2 − q − φ(q)
q
nk−1)+R(Mk−1,u).
Similarly, if we continue, we get
R(Mk+1,u) 6 (1 +
q2 − q − φ(q)
q
nk) + (1 +
q2 − q − φ(q)
q
nk−1)
+ · · · + (1 +
q2 − q − φ(q)
q
nr) + R(Mr,u).
Since nk = qk
−1
q−1 , for k > 2, therefore
R(Mk+1,u)
6 (k−r+1)+
q2−q−φ(q)
q
(
qk−1
q−1
+
qk−1−1
q−1
+· · ·+
qr−1
q−1
)
+R(Mr,u)
6 (k−r+1)+
q2−q−φ(q)
q
(
qk+qk−1+· · ·+qr−(k−r+1)
q−1
)
+R(Mr,u)
6
(k−r+1)φ(q)+(q2−q−φ(q))qr(qk−r+qk−r−1+· · ·+1)
q(q−1)
+R(Mr,u)
252 Zq -s implex codes for an even integer q
6
(k−r+1)(q−1)φ(q)+(q2−q−φ(q))qr(qk−r+1−1)
q(q − 1)2
+R(Mr,u).
If u = k, then
R(Mk+1,k) 6
⌊
(q − 1)φ(q)nk
q
⌋
+ (q − 1 − φ(q))nk + 1 for k > 2.
In the above theorem, if we replace r by u + 1, we get
R(Mk+1,u) 6
(k − u)(q − 1)φ(q) + (q2 − q − φ(q))qu+1(qk−u − 1)
q(q − 1)2
+
(q − 1)φ(q)nu
q
+ (q − 1 − φ(q))nu + 1 for u > 2.
Thus, we have
Corollary 2. For k > u > 2,
R(Mk+1,u) 6
(k − u)(q − 1)φ(q) + (q2 − q − φ(q))qu+1(qk−u − 1)
q(q − 1)2
+
(q − 1)φ(q)nu
q
+ (q − 1 − φ(q))nu + 1.
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Contact information
P. Chella Pandian,
C. Durairajan
Department of Mathematics,
Bharathidasan University,
Tiruchirappalli - 620 024, Tamilnadu, India.
E-Mail(s): chellapandianpc@gmail.com,
cdurai66@rediffmail.com
Web-page(s): http://www.bdu.ac.in/schools/
mathematical_sciences/
mathematics
Received by the editors: 17.07.2013
and in final form 31.07.2013.
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