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In this paper, we develop a theory for constructing cyclic codes of odd length over the ring R = ℤ 4 + u ℤ 4 + v ℤ 4 , where u 2 = v 2 = u v = v u = 0 , which plays an important role in DNA computing. A direct link between the elements of R and the 6 4 codons used in the amino acids of the living organisms is established. Then, we investigate reversible cyclic codes and reversible complement cyclic codes of odd length over R. Moreover, we give some properties of binary images of the codons under the Gray map. Finally, two examples of cyclic codes over R with their minimum Hamming distance will be studied as well. [ABSTRACT FROM AUTHOR]

Let A (G) be the adjacency matrix of a graph G. Let s (v) denote the row entries of A (G) corresponding to the vertex v of G. The Hamming distance between the strings s (u) and s (v) is the number of positions in which s (u) and s (v) differ. In this paper, we study the Hamming distance between the strings generated by the adjacency matrix of subgraph complement of a graph. We also compute sum of Hamming distances between all pairs of strings generated by the adjacency matrix of G ⊕ S. [ABSTRACT FROM AUTHOR]

In this paper, we use a method inspired by the valuation theory to classify completely cyclic codes of length 3 p s over p m + u p m + u 2 p m . Then, we compute their separation parameters which was useful to compute the Hamming distance in [8]. [ABSTRACT FROM AUTHOR]

Let A (n , d) be the size of the maximum binary code of length n and minimum Hamming distance d. A (n , d , w) is defined similarly for binary code with constant weight w. Obviously, finding the value of A (n , d) is equivalent to finding the maximum independent set of the d − 1 th-power of n -dimensional hypercube. Based on this, this paper obtains that A (3 m , 2 m) = 4 , A (m (m + 1) 2 , 2 m − 2 , m) = m + 1 , A (2 m + 1 − 2 , 2 m − 1) = 2 m + 1 , and explores the structure of the maximum code of length 2 m + 2 and minimum Hamming distance 2 m + 1 , where m is an integer. [ABSTRACT FROM AUTHOR]

Motivated by a generalization of the football pool problem, we introduce additive cyclic codes over mixed alphabets of the form where , for distinct primes . We study their algebraic properties, generating sets, and duals. Additionally, we give examples of additive cyclic codes over the alphabet that have best-known or optimal parameters. [ABSTRACT FROM AUTHOR]

Let , where and . In this paper, minimum generating sets of cyclic codes over are given. A necessary and sufficient condition for cyclic codes over to be -free is obtained and a BCH-type bound on the minimum Hamming distance for them is also given. [ABSTRACT FROM AUTHOR]