Z4-Linear Perfect Codes

TL;DR

This paper establishes the existence and classification of a specific family of $Z_4$-linear extended perfect codes with distance 4 for lengths that are powers of two greater than 8, highlighting their diversity in ranks.

Contribution

It proves the existence of exactly $[(k+1)/2]$ mutually nonequivalent $Z_4$-linear extended perfect codes for each applicable length and characterizes their ranks.

Findings

Number of such codes is exactly $[(k+1)/2]$ for each length.

All these codes have distinct ranks.

The codes exist for all lengths $n=2^k > 8$.

Abstract

For every $n = 2^k > 8$ there exist exactly $[(k+1)/2]$ mutually nonequivalent $Z_4$-linear extended perfect codes with distance 4. All these codes have different ranks.