The Subfield Codes of Some [<i>q</i> + 1, 2, <i>q</i>] MDS Codes
Ziling Heng, Cunsheng Ding
Abstract
Recently, subfield codes of geometric codes over large finite fields <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> with dimension 3 and 4 were studied and distance-optimal subfield codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(p)$ </tex-math></inline-formula> were obtained, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=p^{m}$ </tex-math></inline-formula> . The key idea for obtaining very good subfield codes over small fields is to choose very good linear codes over an extension field with small dimension. This paper first presents a general construction of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[q+1, 2, q]$ </tex-math></inline-formula> MDS codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> , and then studies the subfield codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(p)$ </tex-math></inline-formula> of some of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[q+1, 2,q]$ </tex-math></inline-formula> MDS codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(q)$ </tex-math></inline-formula> . Two families of dimension-optimal codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(p)$ </tex-math></inline-formula> are obtained, and several families of nearly optimal codes over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {GF}}(p)$ </tex-math></inline-formula> are produced. Several open problems are also proposed in this paper.