Counting The Generator Matrices of $\mathbb{Z}_{2}\mathbb{Z}_{8}$-Codes

TL;DR

This paper introduces Mixed Generalized Gaussian Numbers (MGN) to count generator matrices of $$-codes, generalizing classical Gaussian numbers and providing new insights and sequences in coding theory.

Contribution

It defines MGN as a new counting formula for $$-codes, generalizing Gaussian numbers, and explores its properties and applications.

Findings

MGN generalizes Gaussian numbers for mixed alphabet codes.

MGN formula recovers known counts for binary and $$-codes.

Provides new integer sequences not in OEIS.

Abstract

In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows generate vector spaces with particular dimension over finite fields. Due to this similarity we name this numbers as Mixed Generalized Gaussian Numbers (MGN). The MGN formula by specialization leads to the well known formula for the number of binary codes and the number of codes over $\mathbb{Z}_8,$ and for additive $\mathbb{Z}_2\mathbb{Z}_4$ codes. Also, we conclude by some properties and examples of the MGN numbers that provide a good source for new number sequences that are not listed in The On-Line Encyclopedia of Integer Sequences.