Abstract
We define a family of codes called twisted Hermitian codes, which are based on Hermitian codes and inspired by the twisted Reed–Solomon codes described by Beelen, Puchinger, and Nielsen. We demonstrate that these new codes can have high-dimensional Schur squares, and we identify a subfamily of twisted Hermitian codes that achieves a Schur square dimension close to that of a random linear code. Twisted Hermitian codes allow one to work over smaller alphabets than those based on Reed–Solomon codes of similar lengths.
Highlights
Reed–Solomon and Hermitian codes are algebraic geometry codes based on the projective line and the Hermitian curve, respectively
We present a new family of codes, called twisted Hermitian codes, whose construction is based on Hermitian codes
We identify a subfamily of the new codes that have Schur squares of dimension close to that expected of a random linear code
Summary
Reed–Solomon and Hermitian codes are algebraic geometry codes based on the projective line and the Hermitian curve, respectively. To define an algebraic geometry code, let X be a smooth, projective, absolutely irreducible curve over a finite field F. To obtain a code of the same dimension whose square is much larger, twisted Reed–Solomon codes were defined by Beelen, Puchinger, and Nielsen [4], drawing upon ideas from the twisted Gabidulin codes of Sheekey [5]. These same ideas serve as inspiration for the recent work [6]. Twisted Hermitian codes can have a large Schur square, as demonstrated by making use of field extensions.
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