On the State Approach Representations of Convolutional Codes over Rings of Modular Integers

In this study, we prove the existence of minimal first-order representations for convolutional codes with the predictable degree property over principal ideal artinian rings. Further, we prove that any such first-order representation leads to an input/state/output representation of the code provided the base ring is local. When the base ring is a finite field, we recover the classical construction, studied in depth by J. Rosenthal and E. V. York. This allows us to construct observable convolutional codes over such rings in the same way as is carried out in classical convolutional coding theory. Furthermore, we prove the minimality of the obtained representations. This completes the study of the existence of input/state/output representations of convolutional codes over rings of modular integers.

Maximum distance separable codes over ℤ2 × ℤ2s

Bilal et al. (Maximum distance separable codes over [Formula: see text] and [Formula: see text] Des. Codes Cryptogr. 61 (2011) 31–40) obtained two upper bounds on minimum distance of codes over rings to the case of [Formula: see text]-additive codes and through these bounds, they introduced two kinds of maximum distance separable codes ([Formula: see text] and [Formula: see text]), the minimum distance of which meets any of those bounds. Also, they completely determined these two types of codes. In this paper, we generalize these facts on [Formula: see text]-additive codes and determine all possible parameters of the [Formula: see text] and [Formula: see text] codes over [Formula: see text].

Construction of Linear Codes over Rings Zm Correcting Double ±1 or ±2 Errors

In this paper a construction of double ±1 and ±2 errors correcting linear optimal and quasi-optimal codes over rings Z5, Z7 and Z9 is presented with the limitation that both errors have the same amplitude in absolute value.

Duality preserving Gray maps for codes over rings

Given a finite ring [Formula: see text] which is a free left module over a subring [Formula: see text] of [Formula: see text], two types of [Formula: see text]-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over [Formula: see text] to codes over [Formula: see text]. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance.

On modules with cyclic socle

The extension problem for linear codes over modules with respect to Hamming weight was already settled in [J. A. Wood, Code equivalence characterizes finite Frobenius rings, Proc. Amer. Math. Soc. 136 (2008) 699–706; Foundations of linear codes defined over finite modules: The extension theorem and MacWilliams identities, in Codes Over Rings, Series on Coding Theory and Cryptology, Vol. 6 (World Scientific, Singapore, 2009), pp. 124–190]. A similar problem arises naturally with respect to symmetrized weight compositions (SWC). In 2009, Wood proved that Frobenius bimodules have the extension property (EP) for SWC. More generally, in [N. ElGarem, N. Megahed and J. A. Wood, The extension theorem with respect to symmetrized weight compositions, in 4th Int. Castle Meeting on Coding Theory and Applications (2014)], it is shown that having a cyclic socle is sufficient for satisfying the property, while the necessity remained an open question. Here, landing in midway, a partial converse is proved. For a (not small) class of finite module alphabets, the cyclic socle is shown necessary to satisfy the EP. The idea is bridging to the case of Hamming weight through a new weight function. Note: All rings are finite with unity, and all modules are finite too. This may be re-emphasized in some statements. The convention for left homomorphisms is that inputs are to the left.