On the Decoding Complexity of Cyclic Codes Up to the BCH Bound

The standard algebraic decoding algorithm of cyclic codes $[n,k,d]$ up to the BCH bound $t$ is very efficient and practical for relatively small $n$ while it becomes unpractical for large $n$ as its computational complexity is $O(nt)$. Aim of this paper is to show how to make this algebraic decoding computationally more efficient: in the case of binary codes, for example, the complexity of the syndrome computation drops from $O(nt)$ to $O(t\sqrt n)$, and that of the error location from $O(nt)$ to at most $\max \{O(t\sqrt n), O(t^2\log(t)\log(n))\}$.Comment: accepted for publication in Proceedings ISIT 2011. IEEE copyrigh

Describing A Cyclic Code by Another Cyclic Code

A new approach to bound the minimum distance of $q$-ary cyclic codes is presented. The connection to the BCH and the Hartmann--Tzeng bound is formulated and it is shown that for several cases an improvement is achieved. We associate a second cyclic code to the original one and bound its minimum distance in terms of parameters of the associated code

Algebraic methods for the distance of cyclic codes

In this thesis we provide known and new results which explain the relationship between the actual minimum distance of cyclic codes, bounds that use only information on the defining sets of cyclic codes to lower bound the distance (root bounds) and bounds that also need the knowledge of the defining sets of all cyclic subcodes (border bounds). We propose a new bound which is provably better of many known bounds and that can be computed in polynomial time with respect to the length of the code. We sketch how to use the generalized Newton identities to give alternative proofs of known bounds. Finally, we use Groebner bases to prove that the optimal root bound can be computed in finite time