We obtain finite and asymptotic Gilbert-Varshamov type bounds for linear codes over finite chain rings with various weights.

In this paper, we propose a class of linear codes and obtain their weight distribution. Some of these codes are almost optimal. Moreover, several classes of constant composition codes(CCCs) are constructed as subcodes of linear codes.

Orthogonal matrices over arbitrary elds are de ned together with their non-square analogs, which are termed row-orthogonal matrices. Antiorthogonal and self-orthogonal square matrices are introduced together with their non-square analogs. The relationships of these matrices to such codes as self-dual codes and linear codes with complementary duals are given.

Orthogonal matrices over arbitrary elds are de ned together with their non-square analogs, which are termed row-orthogonal matrices. Antiorthogonal and self-orthogonal square matrices are introduced together with their non-square analogs. The relationships of these matrices to such codes as self-dual codes and linear codes with complementary duals are given. These relationships are used to obta...

We recap our analysis of e-error locating pairs from last time. For more detail, refer to Lecture 11. We defined u ? v = (u1v1, . . . , unvn) for u, v ∈ Fq , and A ? C = {a ? b|a ∈ A, b ∈ B} for A,B ⊆ Fq . Recall that (A,B) is an e-error locating pair for a linear code C iff 1. A,B are linear codes 2. A ? C ⊆ B 3. dim(A) > e 4. ∆(B) > e 5. ∆(C) > n−∆(A) Finding an (A,B) pair which satisfies pro...

Abstract—There is a known best possible upper bound on the probability of undetected error for linear codes. The [n, k; q] codes with probability of undetected error meeting the bound have support of size k only. In this note, linear codes of full support (= n) are studied. A best possible upper bound on the probability of undetected error for such codes is given, and the codes with probability...

The length function lq(r,R) is the smallest length of a q-ary linear code of covering radius R and codimension r. New upper bounds on lq(r, 2) are obtained for odd r ≥ 3. In particular, using the one-to-one correspondence between linear codes of covering radius 2 and saturating sets in the projective planes over finite fields, we prove that

Let Fq be the finite field with q = p m elements, where p is an odd prime and m is a positive integer. Let u be a positive integer and Tr be the trace function from Fq to Fp. We define a p-ary linear codes CD = {c(a, b) = (Tr(ax+ by))(x,y)∈D : a, b ∈ Fq}, where D = {(x, y) ∈ Fq\{(0, 0)} : Tr(x1 + y u+1) = 0} and N1 = 1 or 2. In this paper, we use Weil sums to investigate weight distributions of...