19 research outputs found
Construction of isodual codes from polycirculant matrices
Double polycirculant codes are introduced here as a generalization of double
circulant codes. When the matrix of the polyshift is a companion matrix of a
trinomial, we show that such a code is isodual, hence formally self-dual.
Numerical examples show that the codes constructed have optimal or
quasi-optimal parameters amongst formally self-dual codes. Self-duality, the
trivial case of isoduality, can only occur over \F_2 in the double circulant
case. Building on an explicit infinite sequence of irreducible trinomials over
\F_2, we show that binary double polycirculant codes are asymptotically good
On the equivalence of linear cyclic and constacyclic codes
We introduce new sufficient conditions for permutation and monomial
equivalence of linear cyclic codes over various finite fields. We recall that
monomial equivalence and isometric equivalence are the same relation for linear
codes over finite fields. A necessary and sufficient condition for the monomial
equivalence of linear cyclic codes through a shift map on their defining set is
also given. Moreover, we provide new algebraic criteria for the monomial
equivalence of constacyclic codes over . Finally, we prove that
if , then all permutation equivalent constacyclic codes of
length over are given by the action of multipliers. The
results of this work allow us to prune the search algorithm for new linear
codes and discover record-breaking linear and quantum codes.Comment: 18 page
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
Group Rings, G-Codes and Constructions of Self-Dual and Formally Self-Dual Codes
The final publication is available at Springer via http://dx.doi.org/10.1007/s10623-017-0440-7We describe G-codes, which are codes that are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a G-code is also a G-code. We give constructions of self-dual and formally self-dual codes in this setting and we improve the existing construction given in [13] by showing that one of the conditions given in the theorem is unnecessary and, moreover, it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes must have an automorphism group that contains G as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative [72, 36, 16] Type II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48. We define quasi-G codes and give a construction of these codes
Polyadic Constacyclic Codes
For any given positive integer , a necessary and sufficient condition for
the existence of Type I -adic constacyclic codes is given. Further, for any
given integer , a necessary and sufficient condition for to be a
multiplier of a Type I polyadic constacyclic code is given. As an application,
some optimal codes from Type I polyadic constacyclic codes, including
generalized Reed-Solomon codes and alternant MDS codes, are constructed.Comment: We provide complete solutions on two basic questions on polyadic
constacyclic cdes, and construct some optimal codes from the polyadic
constacyclic cde