IMPACT: International Journal of Research in Applied, Natural and Social Sciences (IMPACT: IJRANSS) ISSN(E): 23218851; ISSN(P): 23474580 Vol. 2, Issue 1, Jan 2014, 6170 © Impact Journals
ON THE COVERING RADIUS OF SOME CODES OVER R = Z2 + uZ2, WHERE u2 = 0 P. CHELLA PANDIAN & C. DURAIRAJAN Department of Mathematics, Bharathidasan University, Tiruchirappalli, Tamil Nadu, India
ABSTRACT In this correspondence, we give lower and upper bounds on the covering radius of codes over the ring R = Z2 + uZ2 where u2 = 0 with respect to different distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type α and Type β) and their dual and give bounds on the covering radius for MacDonald codes of both types over R.
KEYWORDS: Covering Radius, Codes over Finite Rings, Simplex Codes, Hamming Codes 1 INTRODUCTION In the last decade, there are many researchers doing research on code over finite rings. In particular, codes over Z4, Z2 + uZ2 where u2 = 0 received much attention [1, 2, 3, 4, 5, 9, 11, 12, 14, 16, 17]. The covering radius of binary linear codes were studied [6, 7]. Recently the covering radius of codes over Z4 has been investigated with respect to Lee and Euclidean distances [1, 15]. In 1999, Sole et al gave many upper and lower bounds on the covering radius of a code over Z4 with different distances. In the recent paper [15], the covering radius of some particular codes over Z4 have been investigated. In this correspondence, we consider the ring R = Z2 + uZ2 where u2 = 0. In this paper, we investigate the covering radius of the Simplex codes (both types) and their duals, MacDonald codes and repetition codes over R. We also generalized some of the known bounds in [1]. A linear code C of length n over R is an additive subgroup of Rn. An element of C is called a codeword of C and a generator matrix of C is a matrix whose rows generate C. The Hamming weight wH(x) of a vector x in Rn is the number of nonzero components. The Lee weight for a codeword x = (x1, x2,. . . , xn) is defined by n
wL(x) =
∑
i =1
w L ( x i ) , where
w
L
0 (xi) = 1 2
if if
xi
xi = u
if
The Lee distance between the codewords x and y
∈ Rn is defined as dL (x, y) = wL (x  y). The Euclidean weight
n
is given by the relation wE(x) =
∑w i =1
E ( xi )
xi = 0 = 1 or 1 + u
0
if
xi = 0
4
if if
x i = 1or 1 + u xi = u
, where w ( x ) = 1 E i
The Euclidean distance between the codewords x and y
∈
Rn is defined as dE (x, y) = wE (x  y).
2
A linear Gray map φ from R →Z 2 is defined by φ(x + uy) = (y, x + y), for all x + uy ∈ R. The image φ(C), of a linear code C over R of length n by the Gray map, is a binary code of length 2n with same cardinality [16]. The dual code ⊥
C of C is defined as {x C
⊆
∈
n
R  x. y = 0 for all y
∈
n
C} where x.y =
∑
i =0
xi yi
(mod 2). C is selforthogonal if
C⊥ and C is selfdual if C = C⊥. Two codes are said to be equivalent if one can be obtained from the other by
62
P. Chella Pandian & C. Durairajan
permuting the coordinates or changing the signs of certain coordinates or multiplying nonzero element in a fixed column. Codes differing by only a permutation of coordinates are called permutationequivalent. Any linear code C over R is equivalent to a code with generator matrix G of the form
Ik G = 0 0
A uI k 1
B , uD
(1.1)
where A, B and D are matrices over R. Then the code C contain all codewords [v0, v1 ]G, where v0 is a vector of length k1 over R and v1 is a vector of length k2 over Z2. Thus C contains a total of C are given [n,
4 k1 2 k2
codewords. The parameters of
4k1 2k2 , d] where d represents the minimum distance of C. In[11], we associate to the code C, two binary
codes. The residue code C1 is defined as C1 = {x
∈
n
Z2 ∃ y
∈
n
Z 2 and x + uy
∈
C} and the torsion code
C2 = {x ∈ Z 2  ux ∈ C}. A vector v is a 2linear combination of the vectors v1, v2, . . . , vk if v = l1 v1 + . . . + lk vk with n
li
∈
Z 2 for 1 ≤ i ≤ k. A subset S = { v1, v2, . . . , vk } of C is called a 2basis for C if for each i = 1, 2, ..., k −1, 2vi is a
2linear combination of vi+1, ..., vk , 2vk = 0, C is the 2linear span of S and S is 2linearly independent [18]. The number of elements in a 2basis for C is the 2dimension of C. It is easy to verify that the rows of the matrix
IK0 B = uI k 0 0
A uA uI k 1
B uB uD
(1.2)
form a 2basis for the code C generated by G given in (1.1). A linear code C over R (over Z2) of length n, 2dimension k, minimum distance dH and dL is called an [n, k, dH, dL ] ([n, k, dH ]) or simply an [n, k] code. In this paper, we define the covering radius of codes over R with respect to different distances and in particular study the covering radius α
β
of Simplex codes of type α and β namely, S k and S k and their duals, MacDonald codes and repetition codes over R. Section 2 contains basic results for the covering radius of codes over R. Section 3 determines the covering radius of different types of repetition codes. Section 4 determines the covering radius of Simplex codes and its dual and finally section 5 determines the bounds on the covering radius of MacDonald codes.
2 COVERING RADIUS OF CODES
{
}}.
Let d be the general distance out of various possible distances (such as Hamming, Lee, and Euclidean).
{
min d (c, u) The covering radius of a code C over R with respect to a general distance d is given by rd (C) = max n c∈C u∈R
It is easy to see that rd(C) is the least positive integer rd such that
{
S rd (u ) = v ∈ R n d (u , v ) ≤ rd
} for any u ∈
Rn. The translate u+C = {u + c  c
R n = U S rd (c ) c∈C
∈
where
C} is called the coset of
C where u ∈ Rn. A vector of minimum weight in a coset is called a coset leader. The following propositions are straight forward generalization from [1]. Proposition 2.1 The covering radius of C with respect to the general distance d is the largest minimum weight among all cosets.
On the Covering Radius of Some Codes over R = Z2 + uZ2, Where u2 = 0
63
Proposition 2.2 Let C be a code over R and φ(C) the generalized Gray map image of C. Then rL (C) = rH (φ(C)). Now, we give several lower and upper bounds on the covering radius of codes over R with respect to general weight. The proof of Proposition 2.3 and Theorem 2.4 being similar to the case of Z4 [1], is omitted. Proposition 2.3 (SphereCovering Bounds) For any code C of length n over R,
2
2
s −1n
C
≤
rd ( C )
∑
i= 0
2 s −1n i
We consider the two upper bounds on the covering radius of a code over R with respect to general weight. Let C be a code over R and let s(C⊥) = { i  Ai (C ⊥) ≠ 0, i ≠ 0 } where Ai (C⊥) is the number of codewords of various possible distances i in C⊥. Theorem 2.4 (Delsarte Bound) Let C be a code over R, then rd (C) ≤ s(C⊥). The following result of Mattson [6] is useful for computing covering radius of codes over rings generalized easily from codes over finite fields. Proposition 2.5 (Mattson) If C0 and C1 are codes over R generated by matrices G0 and G1 respectively and if C is the code generated by
0 G= G0
G1 then rd (C) ≤ rd (C0) + rd (C1) and the covering radius of D (concatenation of C0 and C1) satisfy the A
following rd (D) ≥ rd (C0) + rd (C1), for all distances d over R. Since the covering radius of C generated by G = [AB] is greater than or equal to rd (CA) + rd (CB) where CA and CB are codes generated by A and B respectively, this implies rd (D) ≥ rd (C0) + rd (C1) because C1 is a subcode of the code generated [G1A].
3 COVERING RADIUS OF REPETITION CODES A qary repetition code C over a finite field Fq = {α0 = 0, α1 = 1, α2, α3, . . . , αq−1 } is an [n, 1, n] code
{
C = α α ∈ F q } , where
α = (α , α , L , α ) . The covering radius of C is
n(q −1) [13]. Using this, it can be q
seen easily that the covering radius of block (of size n) repetition code [n(q1),1,n(q1)] generated by 6 4 4 7n 4 48 67n8 6 4 7n 48 6 4 7n 48 G = 11 L 1 α 2 α 2 L α 2 α 3 α 3 L α 3 L α q − 1 α q − 1 L α q − 1
is
n ( q − 1) 2 q
since it will be equivalent to a repetition code of
length (q − 1)n. Consider the repetition code over R. There are two types of them of length n viz. unit repetition code Cβ : [n, 2, n, n] generated by Gβ = [11 . . . 1] and zero divisor repetition code Cα : [n, 1, n, 2n] generated by Gα = [uu . . . u]. With respect to the Hamming distance the covering radius of Cβ is n ( q − 1) but covering radius of Cα is n.
q
The following result determines the covering radius with respect Lee distance and Euclidean distance.
64
P. Chella Pandian & C. Durairajan
Theorem 3.1 rL (Cα) = n, rE (Cα) = 2n, rL (Cβ) = n and rE (Cβ) = 3n . 2 Proof. Note that φ(Cα) is a binary repetition code of length 2n hence rL (Cα) = 2n = n. Now by definition 2 n
rE (Cα) = max {d E ( x , C α ) } . Let x n u∈R
other hand if x
n
2 2 6 47 4 86 47 4 8 = uuu L u 000 L 0
∈
R n, then dE (x, 0) = dE (x, u) = 2n. Thus rE (Cα) ≥ 2n. On the 3
∈
n
R has a composition (ω0, ω1, ω2, ω3), where
∑w i=0
i
=n
then dE (x, 0 ) = n − ω0 + 3 ω2 and dE(x, u )
= n − ω2 + 3 ω0. Thus dE (x, Cα) = min{ n − ω0 + 3 ω2, n − ω2 + 3 ω0} ≤ n+ ω0 + ω2 ≤ n + n = 2n. Hence rE (Cα)= 2n. Similar arguments
can
be
used
to
show
that
≤
rE(Cβ)
3n . 2
To
show
that
rE(Cβ)
3n , 2
≥
let
t t t − 3t 6 47 4 8 67 8 67 8 644n7 4 48 x = 000 L 0111 L1uu L u L1 + u1 + u L1 + u ∈ Rn, where t = n , then dE (x, 0 ) = n+2t, dE (x, 1 ) = 4n−10t,
4
dE (x, u ) = n+2t and dE (x, 1 + u ) = 6t. Thus rE (Cβ) ≥ min{4n − 10t, n + 2t, 6t} ≥ 3n . The proof of rL (Cβ) = n is simple so 2
we omit it. In order to determine the covering radius of Simplex and MacDonald codes over R, we need to define few block repetition codes over R and find their covering radius. To determine the covering radius of R block (three blocks each of 3n
size n) repetition code BRep α
n n n 67 8 67 8 64 47 4 48 : [3n, 2, 2n, 4n, 6n] generated by G = [ 111 L 1uu L u L 1 + u1 + u L 1 + u
the code has constant Lee weight 4n. Thus for x = 11 3n
L1 ∈
]
note that
3n
R3n, we have dL (x, BRep α ) = 3n. Hence by definition,
3n
rL (BRep α ) ≥ 3n. On the other hand, its Gray image φ(BRep α ) is equivalent to binary linear code [6n, 2, 4n] with the 2n 8 67 2n 2n 11 L 1 678 6 7 8 3n generator matrix 11 L 1 00 L 0 . Thus the covering radius rL (BRep α ) ≤ 4n + 2n = 3n. Thus, we 2 2 L 11 12 31 00 L 0 11 L 1 2n
3n
3n
have rL (BRep α ) = 3n. With respect Euclidean distance, it is clear that rE (BRep α ) ≥ 3n + 2 n + 3n = 5n. 2
Let x = (uvw)
∈
2
3n
R with u, v and w have compositions (r0, r1, r2, r3), (s0, s1, s2, s3) and (t0, t1, t2, t3) respectively such that
sum of each component composition is n, then dE (x, 0 ) = 3n − r0 + 3 r2 − s0 + 3 s2 − t0 + 3 t3, dE (x, c1) = 3n − r1 + 3 r3 − s2 + 3s0 − t3 + 3 t1, dE (x, c2) = 3n− r2 +3 r0 − s0 +3 s2 − t2 +3 t0 and dE (x, c3) = 3n− r3 +3 r1 −s2+3s0 −t1+3t3. Thus 3n
dE (x, BRep α ) ≤ 3n + min{3r2 + 3 s2 + 3 t2 − r0 − s0− t0, 3 r3 + 3 s0 + 3 t1 − r1 − s2 − t3, 3r0 + 3 s2 + 3t0 − r2 − s0 − t2, 3 r1 + 3 s0 + 3 t3 − r3 − s2 − t1 } ≤ 3n + 1 {2n + 2 s0 + 2s2 } ≤ 5n. Thus we have the following theorem. 2
Theorem 3.2 3n
3n
rL (BRep α ) = 3n and rE (BRep α ) = 5n.
On the Covering Radius of Some Codes over R = Z2 + uZ2, Where u2 = 0
65 2n
One can also define a R block (two blocks each of size n) repetition code Brep α
: [2n, 2, n, 2n, 4n] generated by
67n 8 6 7n 8 . We have following theorem (its proof is similar to the proof of Theorem 3.2) so we omit it. G = 11 L 1 uu L u
Theorem 3.3 2n 2n rL (BRep α ) = 2n and rE (BRep α ) = 7n 2 2n
Block code Brep α BRep
m+ n
can be generalized to a block repetition code (two blocks of size m and n respectively)
: [m + n, 2, m, min{2m, m + 2n}, min{4m, m + 4n}] generated by
m 8 6 7n 8 67 G = 11 L 1 uu L u
. Theorem
3.3 can be easily generalized using similar arguments to the following. Theorem 3.4 rL (BRep m+n) = m + n and rE (BRep m+n) = 2n + 3m 2
4 SIMPLEX CODES OF TYPE α AND β OVER R Quaternary simplex codes of type α and
β
α
have been recently studied in [2]. Type α Simplex code S k is a linear
code over R with parameters [22k, 2k, 22k1, 22k, 3.22k1] and an inductive generator matrix given by
00L 0 11L1 uu L u 1 + u1 + u L1 + u Gkα = α Gkα−1 Gkα−1 Gkα−1 Gk −1 α
α
(4.1)
β
with G 1 =[0 1 u 1+u]. The dual code of S k is a [22k, 22k+1  2k] code. Type simplex code S k is a punctured α
version of S k with parameters [2k1 (2 k  1), 2k, 22(k1), 2(k1) (2 k  1), 2k (3.2k2 1)] and an inductive generator matrix given by
G
β 2
1 = 1
1 1
and for k > 2 G
1 u β k
1 1+ u
0 1
11 L 1 = α G k −1
u 1
00 L 0 G kβ − 1
α
(4.2)
uu L u G kβ − 1
α
(4.3)
β
where G k −1 is the generator matrix of S k −1 . For details the reader is refereed to [2]. The dual code of S k is a [2k1 (2 k  1), 22k  2 k 2k] type α code with minimum Lee weight d L = 3. Theorem 4.1 α
α
22k ≤ r L (S k ) ≤ 22k+1 and rE (S k
)
k ≤ 5. 4 + 5 3
66
P. Chella Pandian & C. Durairajan
Proof. Let x = 11….. 1
α
∈ Rn, we have d L (x, S k
α
) = 22k. By definition, r L (S k ) ≥ 22k. To find the upper bound,
by equation 4.1, the result of Mattson for finite rings and using Theorem 3.2, we get 2 ( k −1 )
2 ( k −1 )
α
r L (S k
2 ( k −1 )
2 2 7448 6 78 6278 644 α ) ≤ r L (S k −1 )+ r L (< 11 L 1 uu L u 1 + u1 + u L1 + u >)
= r L (S αk − 1 )+ 3.22(k1) α
= 3.22(k1) + 3.22(k2) + 3.22(k3) +…..+ 3.22.2 + 3.22.1 + r L (S 1 ) α
α
α
r L (S k ) ≤ 22k+1(since r L (S 1 ) = 5). Thus 22k ≤ r L (S k ) ≤ 22k+1. β α Similar arguments can be used to show rE (S k ) ≤ 5(4(k1) +4(k2) +4(k3) +….+ 41+1)  11 + r E (S 1 ). 2 α k α rE (S k ) ≤ 5.4 + 5 (since rE (S 1 ) = 8). Similar arguments will compute the covering radius of Simplex codes of
3
type
β . We provide an outline of the proof.
Theorem 4.2 β
β
r L (S k ) ≤ 2k  1 (2k  1) – 1 and rE (S k
)
k k ≤ 5 .4 − 6 .2 − 8
6
Proof. By equation 4.3, Proposition 2.5 and Theorem 3.4, we get ( k −1 )
( 2 k −3 )
( k −2)
−2 4 8 6 78 2 67 β β r L (S k ) ≤ r L (S k −1 )+ r L (< 11 L1 uu L u
>)
β
= r L (S k −1 )+ 2(2k  2) + 2(2k  3)  2(k  2) β
≤ (2(2k  2) + 2(2k  4) +. . . +24)+(2(2k  3) + 2(2k 5)+ . . .+ 23)  (2(k  2) + 2(k  3) + . . .+2) + r L (S 2 ) β
β
Thus r L (S k ) ≤ 2k  1 (2k  1)  1(since r L (S 2 ) = 5). Similarly, by using Theorem 3.4, we derive β β r E (S k ) ≤ 3 (4(k  1) + 4(k  2) +. . . +42) + (4(k  1) + 4(k  2) +. . . +42) 2 (2k  2 + 2k  3 + . . . + 2) +rE (S 2 ) 2 β
r E (S k ) ≤
β 5 . 4 k − 6 .2 k − 8 (since r E (S 2 ) = 8). 6
Theorem 4.3
rL ( S kα ⊥ ) = 1, r L ( S kβ ⊥ ) = 2, r E ( S kα ⊥ ) ≤ 4 and r E ( S kβ ⊥ ) ≤ 4.
On the Covering Radius of Some Codes over R = Z2 + uZ2, Where u2 = 0
Proof. By Delsarte Bound, r L ( S
α ⊥
β⊥
≤ 1 and r L ( S k
)
k
67
) ≤ 2. Thus equality follows in the first case.
For second case, note that r L ( S kβ ⊥ ) ≠1, by sphere covering bound. The result for Euclidean distance follows from Delsarte bound.
5 MACDONALD CODES CODES OF TYPE α AND β OVER R The q k − q q − 1
qary
t
MacDonald
k −1
,k,q
− q
t −1
M k ,t ( q )
code
over
the
finite
field
code in which every nonzero codeword has weight either
Fq
q k −1
is
a
unique
or q k −1 − q t −1 [10].
In [8], authors have defined the MacDonald codes over R using the generator matrices of simplex codes. For 1 ≤ t ≤ k – 1, let G
α k ,t
(G
β
α
G
) be the matrix obtained from G
k ,t
(G
t
β
α
) . i. e, G
t
and G kβ, = t
[
G kβ \
[
=
k ,t
α
G
k
α
(G
k ,
β k ,
]
0 G tα
\
) by deleting columns corresponding to the columns of (5.1)
]
0 G tβ
(5.2)
where [A \ B ] denotes the matrix obtained from the matrix A by deleting the columns of the matrix B and 0 in 5.1 (respectively(5.2)) is a (k  t) x 22t (respectively (k  t) x 2t M
G
α k ,t
α k ,t
: [2
(G
2 k
β k ,t
− 2
2 t
, 2 k ]( M
β k ,t
: [( 2
k −1
− 2
t −1
)( 2
k
+ 2
t

1
(2t  1)) zero matrix. The code
− 1 ), 2 k ]) generated
by
the
matrix
) is the punctured code of S kα ( S kβ ) and is called a MacDonald code of type α ( β ). Next Theorem
provides basic bounds on the covering radius of MacDonald codes. Theorem 5.1 α
rL ( M
k ,t
rE ( M
k ,t
α
) ≤ 4 k − 4 r + rL ( M )≤
α r ,t
) for t < r ≤ k,
5 ( 4 k − 4 r ) + rE ( M 3
α
) for t < r ≤ k.
r ,t
Proof. By Theorem 3.2, α
rL ( M
k ,t
) ≤ 3 .2
(2 k − 2 )
+ rL ( M
α
k − 1,t
)
≤ 3 . 2 ( 2 k − 2 ) + 3 . 2 ( 2 k − 4 ) + ... + 3 . 2
= 4 k − 4 r + rL ( M Similar arguments holds for r E ( M following bounds can be obtained for type
β
α k ,t
α r ,t
)
2r
+ rL ( M
α r ,t
.
) . Similarly using equation 5.2, Proposition 2.5 and Theorem 3.4,
MacDonald code.
Theorem 5.2
rL ( M
β k ,t
) , k ≥ r > t.
) ≤ 2 ( k −1 ) ( 2 k − 1) + 2 ( r −1 ) (1 − 2 r ) + rL ( M
β r ,t
) for t < r ≤ k,
68
P. Chella Pandian & C. Durairajan
rE ( M
β k ,t
) ≤
2 k (5 .2
k
− 6 ) + 2 r ( 6 − 5 .2 r ) + rE ( M 6
β r ,t
) for t< r ≤ k
ACKNOWLEDGEMENTS The first author would like to gratefully acknowledge the UGCRGNF [Rajiv Gandhi National Fellowship (No. F. 142(SC)/2010/SAIII)], New Delhi for providing fellowship and the second author was supported by a grant (SR/S4/MS: 588/09) for the Department of Science and Technology, New Delhi.
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In this correspondence, we give lower and upper bounds on the covering radius of codes over the ring R = Z2 + uZ2 whe...