Pisano period codes

TL;DR

This paper studies a special class of cyclic codes derived from Fibonacci recurrences over prime fields, revealing their weight distributions and providing counterexamples to existing classifications.

Contribution

It introduces a new class of Fibonacci-based cyclic codes, showing their weight properties and challenging previous conjectures on irreducible cyclic codes.

Findings

Codes have either one or two weights.

Irreducible codes provide counterexamples to prior conjectures.

Duals of reducible codes are uniformly packed.

Abstract

The cyclic codes with parity check polynomial the reciprocal of the characteristic polynomial of the Fibonacci recurrence over a prime finite field are shown to have either one weight or two weights. When these codes are irreducible cyclic we obtain many counterexamples to the conjectural classification of two-weight irreducible cyclic codes of Schmidt and White (2002). When they are reducible and projective their duals are uniformly packed.