# The lengths of projective triply-even binary codes

## TL;DR

This paper proves the non-existence of binary projective triply-even codes of length 59, completing the classification of possible lengths for such codes and confirming their existence only at specific lengths.

## Contribution

It establishes the final open case for the length of projective triply-even binary codes, providing a complete characterization of their possible lengths.

## Key findings

- No such codes of length 59 exist.
- Codes exist at lengths 15, 16, 30, 31, 32, 45--51, and ≥60.
- The classification of lengths is now complete.

## Abstract

It is shown that there does not exist a binary projective triply-even code of length $59$. This settles the last open length for projective triply-even binary codes. Therefore, projective triply-even binary codes exist precisely for lengths $15$, $16$, $30$, $31$, $32$, $45$--$51$, and $\ge 60$.