Related papers: MacWilliams Identities for $m$-tuple Weight Enumer…
The formal weight enumerators were first introduced by M. Ozeki. They form a ring of invariant polynomials which is similar to that of the weight enumerators of Type II codes. Later, the zeta functions for linear codes were discovered and…
Linear codes have been an interesting topic in both theory and practice for many years. In this paper, for a prime $p$, we determine the explicit complete weight enumerators of a family of linear codes over $\mathbb{F}_p$ with defining set…
Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation…
We give new proofs of MacMahon and Russell's modulo 6 identities using the method of weighted words. We also present a new refinement of MacMahon's identity, some related finite sum identities, and a companion partition theorem to sequence…
We study a new extension of the weak MSO logic, talking about boundedness. Instead of a previously considered quantifier U, expressing the fact that there exist arbitrarily large finite sets satisfying a given property, we consider a…
In this paper, we establish the matroid structures corresponding to data-local and local maximally recoverable codes (MRC). The matroid structures of these codes can be used to determine the associated Tutte polynomial. Greene proved that…
We prove an Assmus-Mattson-type theorem for block codes where the alphabet is the vertex set of a commutative association scheme (say, with $s$ classes). This in particular generalizes the Assmus-Mattson-type theorems for…
We study and derive identities for the multi-variate independence polynomials from the perspective of heaps theory. Using the inversion formula and the combinatorics of partially commutative algebras we show how the multi-variate version of…
We extend Ravagnani's MacWilliams duality theory to the settings of rank metric codes over finite chain rings, relating the sequences of $q$-binomial moments of a rank metric code over this class of rings with those of its dual.
In 1997 Rosenthal and York defined generalized Hamming weights for convolutional codes, by regarding a convolutional code as an infinite dimensional linear code endowed with the Hamming metric. In this paper, we propose a new definition of…
A weight-dependent generalization of the binomial theorem for noncommuting variables is presented. This result extends the well-known binomial theorem for q-commuting variables by a generic weight function depending on two integers. For a…
Rogers-Ramanujan type identities occur in various branches of mathematics and physics. As a classic and powerful tool to deal with Rogers-Ramanujan type identities, the theory of Bailey's lemma has been extensively studied and generalized.…
In the present paper, we give harmonic weight enumerators and Jacobi polynomials for the first-order Reed--Muller codes and the extended Hamming codes. As a corollary, we show the nonexistence of combinatorial $4$-designs in these codes.
In this paper, we define some weighted sums of the alternating multiple $T$-values (AMTVs), and study several duality formulas for them by using the tools developed in our previous papers. Then we introduce the alternating version of the…
Determining the weight distribution of a code is an old and fundamental topic in coding theory that has been thoroughly studied. In 1977, Helleseth, Kl{\o}ve, and Mykkeltveit presented a weight enumerator polynomial of the lifted code over…
In 1970 Delsarte, Goethals and Mac Williams published a seminal paper on generalized Reed-Muller codes where, among many important results, they proved that the minimal weight codewords of these codes are obtained through the evaluation of…
Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight…
To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of…
We establish a connection between linear codes and hyperplane arrangements using the Thomas decomposition of polynomial systems and the resulting counting polynomial. This yields both a generalization and a refinement of the weight…
Based on a bijection due to Fu and Tang, we provide combinatorial proofs of several partition identities of Andrews and Merca. We also introduce two weights for partitions to extend one of these identities.