Quasicyclic Codes of Index ℓ over F q Viewed as F q[x]-Submodules of F q ℓ[x]/〈x m−1〉

Algebraic lower bounds on the free distance convolutional codes

“…A similar idea was previously used by Séguin [17] to describe convolutional codes when divides , and by this author [9] and more recently by Séguin [18] to describe quasi-cyclic codes. Using our new module approach we establish further links between convolutional codes and cyclic and quasi-cyclic codes.…”

“…As described in [9], is the annihilator ideal of the -module and so every quasi-cyclic code can also be viewed as an -submodule of . As such the code is a subset subcode of the cyclic code over generated as an -submodule of (an ideal in ) by the same set of generators.…”

“…It has recently been shown in [9] that a quasi-cyclic code of index and length can be viewed as an -submodule of . In this setting the rows of a generator matrix of a quasi-cyclic code are successive powers of multiplied modulo by each of the generating polynomials of the submodule in .…”

“…Our new module structure allows us to extend this connection between convolutional and block codes and establish further links to cyclic codes. Results derived in [9] for quasi-cyclic codes can then be adapted and applied to convolutional codes to develop more constructive lower bounds on , in particular a BCH-type bound derived from an associated cyclic code over an extension field. To achieve these additional results some aspects of our alternative proof of the following theorem are required in subsequent sections, and thus included here.…”

Provably Good Codes for Hash Function Design

“…Further, the above codes have the property that they are closed under column rotations. Such codes are called quasi-cyclic codes in the literature, and have been studied extensively (see [13,4,8,9]). As for estimating the minimum weight of such codes by algorithmic means, the presently known techniques are computationally infeasible [4,8].…”