Weight Enumerators and Cardinalities for Number-Theoretic Codes

Distance Enumerators for Number-Theoretic Codes

The number-theoretic codes are a class of codes defined by single or multiple congruences and are mainly used for correcting insertion and deletion errors. Since the number-theoretic codes are generally non-linear, the analysis method for such codes is not established enough. The distance enumerator of a code is a unary polynomial whose $i$th coefficient gives the number of the pairs of codewords with distance $i$. The distance enumerator gives the maximum likelihood decoding error probability of the code. This paper presents an identity of the distance enumerators for the number-theoretic cod…

用不变量理论研究自对偶量子码的三种重量枚举式

An invariant-theoretic approach to three weight enumerators of self-dual quantum codes

本文基于不变量理论研究自对偶量子码的三种重量枚举式，推导出量子格莱森定理的类比，并计算了特殊自对偶量子码的完整重量枚举式。

Explicit Formulas for the Weight Enumerators of Some Classes of Deletion Correcting Codes

We introduce a general class of codes which includes several well-known classes of deletion/insertion correcting codes as special cases. For example, the Helberg code, the Levenshtein code, the Varshamov--Tenengolts code, and most variants of these codes including most of those which have been recently used in studying DNA-based data storage systems are all special cases of our code. Then, using a number theoretic method, we give an explicit formula for the weight enumerator of our code which in turn gives explicit formulas for the weight enumerators and so the sizes of all the aforementioned …

On the Classification of Type II Codes of Length 24

We give a new, purely coding-theoretic proof of Koch's criterion on the tetrad systems of Type II codes of length 24 using the theory of harmonic weight enumerators. This approach is inspired by Venkov's approach to the classification of the root systems of Type II lattices in R^{24}, and gives a new instance of the analogy between lattices and codes.