arXiv subjects

Somphong Jitman

Publications and source records attributed to Somphong Jitman.

39 records · Page 3Linked to original sources

arXiv2010

Checkable Codes from Group Rings

We study codes with a single check element derived from group rings, namely, checkable codes. The notion of a code-checkable group ring is introduced. Necessary and sufficient conditions for a group ring to be code-checkable are given in the case where the group is a finite abelian group and the ring is a finite field. This characterization leads to many good examples, among which two checkable codes and two shortened codes have minimum distance better than the lower bound given in Grassl's online table. Furthermore, when a group ring is code-checkable, it is shown that every code in such a group ring admits a generator, and that its dual is also generated by an element which may be deduced directly from a check element of the original code. These are analogous to the generator and parity-check polynomials of cyclic codes. In addition, the structures of reversible and complementary dual checkable codes are established as generalizations of reversible and complementary dual cyclic codes.

cs.IT↗

arXiv2010

Skew Constacyclic Codes over Finite Chain Rings

Skew polynomial rings over finite fields ([7] and [10]) and over Galois rings ([8]) have been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-λ$, where $λ$ is a unit element, are exhibited. When $λ^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Of more interest are codes over the ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. The structure of all skew constacyclic codes is completely determined. This allows us to express generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length~2 over $\mathbb{F}_{3}+u\mathbb{F}_{3}$ and their Euclidean and Hermitian dual codes is also provided.

cs.IT↗

arXiv2009

The Gray Image of Codes over Finite Chain Rings

The results of J. F. Qiann et al. [4] on $(1-γ)$-cyclic codes over finite chain rings of nilpotency index 2 are extended to $(1-γ^e)$-cyclic codes over finite chain rings of arbitrary nilpotency index $e+1$. The Gray map is introduced for this type of rings. We prove that the Gray image of a linear $(1 - γ^{e})$-cyclic code over a finite chain ring is a distance-invariant quasi-cyclic code over its residue field. When the length of codes and the characteristic of a ring are relatively prime, the Gray images of a linear cyclic code and a linear $(1+γ^e)$-cyclic code are permutatively to quasi-cyclic codes over its residue field.

math.RA↗