Abstract. We present new codes by perturbation of rows of the generating matrix of a given linear code. Some properties of the perturbed linear codes are given

Lalit K Vashisht

Pankaj K Das

CiteSeerX

Math. Appl. 4 (2015), 91–99
DOI: 10.13164/ma.2015.07
ON PERTURBATION OF BINARY LINEAR CODES
PANKAJ K. DAS and LALIT K. VASHISHT
Abstract. We present new codes by perturbation of rows of the generating matrix
of a given linear code. Some properties of the perturbed linear codes are given.
1. Introduction and preliminaries
In coding theory, many methods from elementary to more complicated ones are
used to construct new codes from one or more given codes. Some examples of
such new codes are product code [3], punctured code [6], shortened code [4], and
extended code [4]. The new codes are developed in order to obtain a better code
in some sense or other. Akavia and Venkatesan [1] presented a new class of pertur-
bation codes that are obtained from old codes using perturbation operator. They
analyze the rate and distance of perturbation codes.
In this paper, we discuss a type of perturbation in which the rows vi of a gen-
erator matrix G of an (n, k) linear code C are perturbed by a non-zero vector,
i.e.,
G+r u ≡

v1 + u
v2 + u
...
vk + u
 ,
where u is a non-zero vector which may or may not be in C. In general, G +r u
does not form a generator matrix of the given linear code. It would be inter-
esting to know whether, the perturbed system G +r u forms a generator matrix
of a linear subcode or under which conditions it generates the original code. In
this direction we give necessary and sufficient conditions for the perturbation of
linear codes for generating new (or original) linear codes. A decomposition theo-
rem and MacWilliams type identity for the perturbed linear code are given. An
error detection and error correction relation between a given linear code and the
corresponding perturbed code have been discussed.
Now we give some basic definitions which will be used throughout this paper.
The set V n denotes the space of all n-tuples over a finite field GF (q) (q is a power
of some prime number) with the usual inner product 〈., .〉. Two vectors x and y in
MSC (2010): primary 94B05; secondary 94B20, 94B65.
Keywords: linear codes, generator matrix, perturbation.
The second author is partly supported by R & D Doctoral Research Programme, University
of Delhi, Delhi-110007, India (Grant No.: RC/2014/6820).
91
92 P.K. DAS and L.K. VASHISHT
V n are said to be orthogonal if 〈x, y〉 = 0. Any proper subspace C of V n is called
a linear code and the elements of C are called code words (or code vectors). A
subspace of V n with every element being orthogonal to every element of C is also
a linear code and known as the dual code of C denoted by C⊥. If the dimension of
C is k, then C is called an (n, k) linear code. A k×n matrix whose rows span C is
called a generator matrix. Any generator matrix H of C⊥ is called a parity check
matrix of C. An (n, k) linear code C over a finite field GF (q) is called a cyclic
code if, for every code word v = (v1, v2, . . . , vn) of C, the word (vn, v1, . . . , vn−1)
obtained by a cyclic shift of components is again a code word.
The Hamming weight of a vector is the number of its non-zero components. The
Hamming distance between two vectors is the number of components in which they
differ. The minimum distance for a linear code equals the minimum weight of its
non-zero vectors. The distance of a linear code is the minimum distance of the
code.
Recall that, for sets U and W , U ⊕W denote the subspace consisting of all
linear combinations ax+ by where x ∈ U, y ∈W and a, b are scalars. For a set E,
|E| denotes the number of elements in E.
2. Perturbation of linear codes
In this section, we perturb a generator matrix of a linear code by a non-zero vector.
First we take a non-zero element from the code and call this operation on linear
codes an inner perturbation of linear codes. The following proposition gives the
necessary condition for an inner perturbation of linear codes.
Proposition 2.1. Let C be a binary (n, k) linear code generated by a matrix
G =

v1
v2
...
vk
 .
If each row vi of G is perturbed by the linear sum of the form u =
∑k
i=1 vi, then
the perturbed matrix
G0 =

v1 + u
v2 + u
...
vk + u
 (2.1)
gives rise to the same linear code C provided that k is even.
Proof. It is sufficient to show that the rows in matrix G0 are linearly indepen-
dent. Assume that
α1(v1 + u) + α2(v2 + u) + α3(v3 + u) + · · ·+ αk(vk + u) = 0 (2.2)
for some scalars αi ∈ GF (2) (1 ≤ i ≤ k). Now
α1(v1 + u) + α2(v2 + u) + α3(v3 + u) + · · ·+ αk(vk + u)
= (α2 + α3 + · · ·+ αk)v1 + (α1 + α3 + · · ·+ αk)v2 + . . .
PERTURBATION OF LINEAR CODES 93
+ (α1 + α2 + · · ·+ αk−1)vk.
Since vi’s are linearly independent, by (2.2), we have
α2 + α3 + · · ·+ αk = 0,
α1 + α3 + · · ·+ αk = 0,
. . .
. . .
α1 + α2 + · · ·+ αk−1 = 0.
This yields α1 = αi (2 ≤ i ≤ k). Thus, (k − 1)αi = 0 (1 ≤ i ≤ k). If k even, then
αi = 0. The proposition is proved. 
Remark 2.2. The result given in Proposition 2.1 is not true for odd values of
k. Indeed, let C be a (6, 3) linear code with generator matrix
G =
1 1 0 1 0 00 1 1 0 1 0
0 0 1 1 0 1
 .
Choose u = (100011)
(
=
∑3
i=1 vi
)
. Then, the perturbed matrix (in the sense of
(2.1)) is given by
G0 =
v1 + uv2 + u
v3 + u
 =
0 1 0 1 1 11 1 1 0 0 1
1 0 1 1 1 0
 .
One may observe that the sum of the first two rows in G0 is equal to the third row
in G0. Hence, the perturbed matrix G0 cannot be a generator matrix of the given
code C. More precisely, the idea given in the above example can be generalized to
any (n, k) linear code where n is arbitrary and k is odd.
The following proposition provides a subcode with distance increased by an
inner perturbation.
Proposition 2.3. Assume that d is the distance of an (n, k) binary cyclic linear
code C generated by the matrix
G =

v1
v2
...
vk
 where vj is cyclic shift of vj−1 for all j with 2 ≤ j ≤ k.
If the weight of v1 = d and d is odd, then the perturbed matrix
G0 =

v1 + u
v2 + u
. . .
vi−1 + u
vi+1 + u
. . .
vk + u

where u = vi for some i with 1 ≤ i ≤ k, (2.3)
94 P.K. DAS and L.K. VASHISHT
generates an (n, k− 1) binary linear subcode of a distance greater than or equal to
d+ 1.
Proof. First we note that the rows of matrix G0 (see (2.3)) are linearly inde-
pendent, so G0 generates an (n, k − 1) linear subcode C1 of C. Since all the row
code vectors in G are of odd weight, therefore, all the row code vectors in G0 are of
an even weight. Also, the linear sum of vectors of even weight will produce vector
of an even weight. Hence, all code vectors of C1 are of even weights. Since the
minimum weight of C is d which is odd and C1 is a subcode of C, the minimum
distance of C1 ≥ d+ 1. 
The next result provides a necessary condition for a perturbation of a given
linear code in terms of an eigenvalue of a certain matrix.
Theorem 2.4. Assume that C is a binary (n, k) linear code generated by a ma-
trix
G =

v1
v2
...
vk
 .
Let z1, z2, . . . , zm be linearly independent vector in C and let, for each j (1 ≤ j ≤
m), xj be vectors (not necessarily code vectors) with binary components and let
α
(i)
j be scalars given by α
(i)
j = 〈vi, xj〉 for all i with 1 ≤ i ≤ k. If the perturbed
matrix
Gp =

v1 +
∑m
j=1 α
(1)
j zj
v2 +
∑m
j=1 α
(2)
j zj
...
vk +
∑m
j=1 α
(k)
j zj

is a generator of C, then λ = ±1 is not an eigenvalue of the matrix
L =

〈z1, x1〉 〈z1, x2〉 . . . 〈z1, xm〉
〈z2, x1〉 〈z2, x2〉 . . . 〈z2, xm〉
...
...
. . .
...
〈zm, x1〉 〈zm, x2〉 . . . 〈zm, xm〉
 .
Proof. We prove the theorem for the case m = 2 and λ = 1. The other cases
can be proved similarly. Assume that 1 is an eigenvalue of the matrix
J2 =
[
〈z1, x1〉 〈z1, x2〉
〈z2, x1〉 〈z2, x2〉
]
.
Then ∣∣∣∣〈z1, x1〉 − 1 〈z1, x2〉〈z2, x1〉 〈z2, x2〉 − 1
∣∣∣∣ = 0.
Therefore, we can find scalars α and β not both zero (α, β ∈ GF (2)) such that
α〈z1, x1〉+ β〈z1, x2〉 = α
PERTURBATION OF LINEAR CODES 95
and
α〈z2, x1〉+ β〈z2, x2〉 = β.
Put z = αx1 + βx2. Then, z is a non-zero vector.
By using
〈vi, z〉 = 〈vi, αx1 + βx2〉 = αα(i)1 + βα
(i)
2 for all i (1 ≤ i ≤ k),
we compute
〈vi +
2∑
j=1
α
(i)
j zj , z〉 = 〈vi, z〉+ α
(i)
1 〈z1, z〉+ α
(i)
2 〈z2, z〉
= (αα
(i)
1 + βα
(i)
2 ) + α
(i)
1 α+ α
(i)
2 β
= 0, (1 ≤ i ≤ k). (2.4)
By the hypothesis, Gp is the generator matrix of the given code C. Therefore, the
zero vector is only code vector which is orthogonal to all rows that appear in Gp.
Hence, by using (2.4), we conclude that z = 0, a contradiction. Therefore, 1 is not
an eigenvalue of matrix J2. 
Application of Theorem 2.4: Consider the generator matrix G of a linear
(7, 3) code C:
G =
1 0 1 0 1 0 00 1 0 1 0 1 0
0 0 1 0 1 0 1
 =
v1v2
v3
 .
Choose z1 = x1 = v1; z2 = x2 = v2. Then,
〈z1, x1〉 = 1, 〈z1, x2〉 = 0; 〈z2, x1〉 = 0, 〈z2, x2〉 = 1.
Therefore, −1 is an eigenvalue of the matrix
J2 =
[
〈z1, x1〉 〈z1, x2〉
〈z2, x1〉 〈z2, x2〉
]
.
Hence, by Theorem 2.4 the matrix
Gp =
v1 +
∑2
k=1 α
(1)
k zk
v2 +
∑2
k=1 α
(2)
k zk
v3 +
∑2
k=1 α
(3)
k zk

is not a generator of the given code C.
The case of outer perturbation: Now we discuss the case G +r u where
u /∈ C. First, we give a result which gives a link between the cardinality of a given
linear code and the corresponding outer perturbed code.
Proposition 2.5. Let C be a binary (n, k) linear code generated by a matrix
G =

v1
v2
...
vk
 .
96 P.K. DAS and L.K. VASHISHT
Let u be a non-zero vector which is not in C and let C ′ be the binary linear code
generated by the perturbed matrix
Gu =

v1 + u
v2 + u
...
vk + u
 .
Then, |C
⋂
C ′| = 12 |C| =
1
2 |C
′|.
Proof. By definition of Gu, one may observe that∑
even number of terms
(vj + u) =
∑
even number of terms
vj ∈ C (2.5)
and ∑
odd number of terms
(vj + u) =
( ∑
odd number of terms
vj
)
+ u /∈ C. (2.6)
Note that the number of ways in which the linear sums in (2.5) can be chosen is(
k
0
)
+
(
k
2
)
+
(
k
4
)
+ · · ·+
(
k
l
)
, (2.7)
where l = k (or k − 1) according to whether k is even or odd.
Similarly, the linear sums in (2.6) can be chosen in(
k
1
)
+
(
k
3
)
+ · · ·+
(
k
t
)
(2.8)
ways where t = k (or k − 1) according to whether k is odd or even.
Since (
k
0
)
+
(
k
1
)
+
(
k
2
)
+
(
k
3
)
+ · · ·+
(
k
k
)
= 2k
and (
k
0
)
+
(
k
2
)
+
(
k
4
)
+ · · ·+
(
k
l
)
=
(
k
1
)
+
(
k
3
)
+
(
k
5
)
+ · · ·+
(
k
t
)
,
where l and t are as above. Therefore, by using (2.7) and (2.8), we have(
k
0
)
+
(
k
2
)
+
(
k
4
)
+ · · ·+
(
k
l
)
= 2k−1.
Therefore, the number of elements which are common to both C and C ′ is equal
to 2k−1. Hence |C
⋂
C ′| = 12 |C| =
1
2 |C
′|. 
Remark 2.6. One may observe that the doping vector u in Proposition 2.5 is
not a code vector of C ′.
Remark 2.7. One can verify the symmetric relation:
(C ′ + u)
⋃
C ′ = (C + u)
⋃
C.
PERTURBATION OF LINEAR CODES 97
Remark 2.8. The dimension of (C⊕C ′) is k+1. Note that dim(C
⋂
C ′) = k−1
and dimC + dimC ′ = 2k. By using the relation
dim(C
⋂
C ′) + dim(C ⊕ C ′) = dimC + dimC ′,
we have dim(C ⊕ C ′) = k + 1.
The following theorem provides a decomposition of an outer perturbed code in
terms of a linear subcode and a nonlinear code.
Theorem 2.9. (Decomposition Theorem) Let C ′ be the binary (n, k) linear
code given in Proposition 2.5. Then C ′ can be decomposed as an (n, k − 1) linear
code A and a nonlinear code B.
Proof. Choose
A =
∑
even number of terms
(vj + u) =
∑
even number of terms
vj ,
and
B =
∑
odd number of terms
(vj + u).
Then, C ′ = A
⋃
B and A
⋂
B = ∅.
To show that A is an (n, k − 1) linear code, it is sufficient to show that the
matrix G0 generates A where
G0 =

v1 + v2
v2 + v3
...
vk−1 + vk
 .
Let α1(v1 + v2) + α2(v1 + v2) + · · · + αk−1(vk−1 + vk) = 0, where αi ∈ GF (2).
Then, α1v1 + (α1 + α2)v2 + · · ·+ (αk−2 + αk−1)vk−1 + αk−1vk = 0. By using the
linear independence of vi, we have αj = 0, for all j = 1, 2, . . . , k− 1. Furthermore,
by the nature of the construction of A, we observe that G0 spans A. Hence, G0 is
a generator matrix of A. Thus, A is an (n, k − 1) linear code. One may observe
that B is a nonlinear code. The theorem is proved. 
The following proposition gives a relation of the weight distribution between the
subspace A of C ′ and the null spaces C⊥( or C ′
⊥
). We use certain ideas developed
in [2].
Proposition 2.10. Assume that A
⋃
B is the decomposition of C ′ given in
Theorem 2.9. Let Ai and Bi be the number of vectors of weight i in A and the
null space C⊥( or C ′
⊥
), respectively. Then
n∑
i=0
Ai
(
n− i
m
)
≥ 2k−1−m
n∑
j=0
Bj
(
n− j
n−m
)
, m = 1, 2, . . . n.
98 P.K. DAS and L.K. VASHISHT
Proof. Consider the (n, k − 1) linear code A. Applying MacWilliams identity
(see Theorem 3.14 in [5]) to the code A, the weight enumerator of A is given by
n∑
i=0
Ai
(
n− i
m
)
= 2k−1−m
n∑
j=0
Dj
(
n− j
n−m
)
, m = 1, 2, . . . n, (2.9)
where Di is the total number of vectors of weight i in A
⊥.
Now A ⊂ C(or C ′), i.e., C⊥ (or C ′⊥) ⊂ A⊥. Thus, Bi ≤ Di. Therefore, by
using (2.9), we have
n∑
i=0
Ai
(
n− i
m
)
≥ 2k−1−m
n∑
j=0
Bj
(
n− j
n−m
)
.

Correction and detection of errors: Now we discuss the error detection
and error correction relation between a given linear code and the corresponding
outer perturbed code.
Proposition 2.11. If C is a binary (n, k) cyclic linear code, then A (where A
is given in the proof of Theorem 2.9) can detect any burst of length n − k + 1 or
less.
Proof. First we show that A is an (n, k − 1) cyclic linear code. Let C be
generated by the generator polynomial g(x). By Theorem 2.9, the generator matrix
G0 of A is given by
G0 =

g(x) + xg(x)
xg(x) + x2g(x)
...
xk−1g(x) + xkg(x)

=

(1 + x)g(x)
x(1 + x)g(x)
...
xk−2(1 + x)g(x)
 .
This shows that A is generated by (1 + x)g(x) and, hence, A is a binary (n, k− 1)
cyclic code.
Since an (n, k) cyclic code detects all burst of length n− k or less, see Theorem
8.5 in [5, p. 229], A detects all burst of length n− k + 1 or less. 
To conclude the paper, we will prove a result regarding the perturbation in the
correction of an error vector.
Proposition 2.12. Let C be a binary (n, k) linear code which can correct error
sets E and E + u (= {t + u : t ∈ E}). Then, C ′ also corrects error sets E and
E + u.
PERTURBATION OF LINEAR CODES 99
Proof. Let e1, e2 be any two error vectors in E or of the form t+ u (t ∈ E). To
prove that C ′ corrects error sets E and E + u, it is sufficient to show that e1 and
e2 are members of two different cosets of C
′. Let, if possible, e1, e2 be members
of the same coset of C ′. Then, e1 − e2 must be a code vector of C ′. Therefore,
e1 − e2 ∈ A or B.
If e1 − e2 ∈ A, then A+ e1 = A+ e2. This means that (C + e1)
⋂
(C + e2) 6= φ,
i.e., C + e1 = C + e2. By using the fact that e1 ∈ C + e1 and e2 ∈ C + e2 and
that C corrects e1 and e2, we have C + e1 6= C + e2. This is a contradiction. If
e1− e2 ∈ B, then e1− e2 +u ∈ C. That is, C + e1 = C + e2 +u. As C corrects e1
and e2+u and that e1 ∈ C+e1 and e2+u ∈ C+e2+u, then C+e1 = C+e2+u is
not possible. Hence, e1 and e2 must be members of two different cosets of C
′. 
Acknowledgment. The authors would like to thank the anonymous reviewer
for his/her careful reading of the paper and valuable comments and suggestions
to improve the quality of the paper.
References
[1] A. Akavia and R. Venkatesan, Perturbation codes, Presented at IPAM Workshop on Locally
Decodable Codes, 2006.
[2] E. F. Assmus (Jr.) and H. F. Mattson (Jr.), The weight-distribution of a coset of a linear
code, IEEE Trans. Inf. Theory 24 (1978), 497–497.
[3] P. Elias, Error-free coding, Information Theory, Transactions of the IRE Professional Group
on 4 (1954), 29–37.
[4] S. J.K. Kenneth, Construction of binary linear codes, Undergraduate Research Opportu-
nity Programme in Science, Department of Mathematics, National University of Singapore,
1999/2000 (www.math.nus.edu.sg/ urops/Projects/BinaryCodes)
[5] W.W. Peterson and E. J. Weldon (Jr.), Error-Correcting Codes, 2nd ed., The MIT Press,
1972.
[6] H. van Tilborg, Coding Theory: A First Course, Eindhoven the Netherlands, 1993.
Pankaj K. Das, Department of Mathematics, Shivaji College, University of Delhi, Raja Gar-
den, New Delhi–110027, India (Presently on lien in Department of Mathematical Sciences, Tezpur
University, Napaam, Sonitpur, Assam–784 028, India)
e-mail : pankaj4thapril@yahoo.co.in, pankaj4@tezu.ernet.in
Lalit K. Vashisht, Department of Mathematics, University of Delhi, Delhi–110007, India
e-mail : lalitkvashisht@gmail.com



ON PERTURBATION OF BINARY LINEAR CODES

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ON PERTURBATION OF BINARY LINEAR CODES

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