Related papers: A duality for nonabelian group codes
We introduce a general class of regular weight functions on finite abelian groups, and study the combinatorics, the duality theory, and the metric properties of codes endowed with such functions. The weights are obtained by composing a…
A MacWilliams Identity for convolutional codes will be established. It makes use of the weight adjacency matrices of the code and its dual, based on state space realizations (the controller canonical form) of the codes in question. The…
A conceptual framework involving partition functions of normal factor graphs is introduced, paralleling a similar recent development by Al-Bashabsheh and Mao. The partition functions of dual normal factor graphs are shown to be a Fourier…
Motivated by the works of Shiromoto [3] and Shi et al. [4], we study the existence of MacWilliams type identities with respect to Lee and Euclidean weight enumerators for linear codes over $\mathbb{Z}_{\ell}.$ Necessary and sufficient…
In this paper, we define dual codes over arbitrary finite rings with respect to arbitrary bilinear forms and provide a generalization of Hayden's theorem (Bridges, Hall, and Hayden, 1981). Building on this foundation, we introduce the…
In this paper, we have introduced the concepts of support distribution and the support enumerator as refinements of the classical weight distribution and weight enumerator respectively, capturing coordinate level activity in linear block…
We initiate the study of the duality theory of locally recoverable codes, with a focus on the applications. We characterize the locality of a code in terms of the dual code, and introduce a class of invariants that refine the classical…
Since MacWilliams proved the original identity relating the Hamming weight enumerator of a linear code to the weight enumerator of its dual code there have been many different generalizations, leading to the development of $m$-tuple support…
Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on…
In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators. The…
The adjacency matrix associated with a convolutional code collects in a detailed manner information about the weight distribution of the code. A MacWilliams Identity Conjecture, stating that the adjacency matrix of a code fully determines…
The MacWilliams Identity is a well established theorem relating the weight enumerator of a code to the weight enumerator of its dual. The ability to use a known weight enumerator to generate the weight enumerator of another through a simple…
The choice of an isomorphism, a duality, between a finite abelian group $A$ and its character group allows one to define dual codes of additive codes over $A$. Properties of dualities and dual codes are studied, continuing work of Delsarte…
Continuing previous works on MacWilliams theory over codes and lattices, a generalization of the MacWilliams theory over $\mathbb{Z}_k$ for $m$ codes is established, and the complete weight enumerator MacWilliams identity also holds for…
One of the most remarkable theorems in coding theory is Gleason's 1970 theorem about the weight enumerators of self-dual codes. In the past 36 years there have been hundreds of papers written about generalizations and applications of this…
In the 1960s, MacWilliams proved that the Hamming weight enumerator of a linear code over a finite field completely determines, and is determined by, the Hamming weight enumerator of its dual code. In particular, if two linear codes have…
We derive a relationship between two different notions of fidelity (entanglement fidelity and average fidelity) for a completely depolarizing quantum channel. This relationship gives rise to a quantum analog of the MacWilliams identities in…
The MacWilliams identity, which relates the weight distribution of a code to the weight distribution of its dual code, is useful in determining the weight distribution of codes. In this paper, we derive the MacWilliams identity for linear…
The MacWilliams identity for linear time-invariant convolutional codes that has recently been found by Gluesing-Luerssen and Schneider is proved concisely, and generalized to arbitrary group codes on graphs. A similar development yields a…
In this paper, we consider codes over finite fields, finite abelian groups, and finite Frobenius rings. For such codes, the complete weight enumerator and the Hamming weight enumerator serve as powerful tools. These two types of weight…