The asymptotic number of binary codes and binary matroids

TL;DR

This paper determines the asymptotic count of nonequivalent binary n-codes and nonisomorphic binary n-matroids, revealing most have trivial automorphism groups, and explores related connections to prior results.

Contribution

It provides the first asymptotic enumeration of binary codes and matroids, linking these counts to automorphism group properties and prior theoretical results.

Findings

Asymptotic number of binary n-codes determined

Most binary n-codes have trivial automorphism groups

Establishes connection to Lefmann, R"odl, Phelps result

Abstract

The asyptotic number of nonequivalent binary n-codes is determined. This is also the asymptotic number of nonisomorphic binary n-matroids. The connection to a result of Lefmann, Roedl, Phelps is explored. The latter states that almost all binary n-codes have a trivial automorphism group.