More results on hulls of some primitive binary and ternary BCH codes

Yinzhao Lei, Chengju Li, Yansheng Wu, Peng Zeng

Abstract: The (Euclidean) hull of a linear code is defined to be the intersection of the code and its Euclidean dual. It is clear that the hulls are self-orthogonal codes, which are an important type of linear codes due to their wide applications in communication and cryptography. Let C<math><mi mathvariant="script" is="true">C</mi></math> be an [n,k]<math><mo stretchy="false" is="true">[</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">k</mi><mo stretchy="false" is="true">]</mo></math> cyclic code over Fq<math><msub is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">F</mi></mrow><mrow is="true"><mi is="true">q</mi></mrow></msub></math>, where Fq<math><msub is="true"><mrow is="true"><mi mathvariant="double-struck" is="true">F</mi></mrow><mrow is="true"><mi is="true">q</mi></mrow></msub></math> is the finite field of order q. In this paper, we will employ the defining set of the code C<math><mi mathvariant="script" is="true">C</mi></math> to present a general characterization when its hull has dimension k−ℓ<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mi is="true">ℓ</mi></math>. Furthermore, we mainly focus on the primitive q-ary BCH codes C(q,n,δ,b)<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">C</mi></mrow><mrow is="true"><mo stretchy="false" is="true">(</mo><mi is="true">q</mi><mo is="true">,</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">δ</mi><mo is="true">,</mo><mi is="true">b</mi><mo stretchy="false" is="true">)</mo></mrow></msub></math> when b=0<math><mi is="true">b</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">0</mn></math> and b=1<math><mi is="true">b</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">1</mn></math> based on the general characterization. Especially for binary and ternary cases, we will present several sufficient and necessary conditions that the hulls of the codes C(q,n,δ,b)<math><msub is="true"><mrow is="true"><mi mathvariant="script" is="true">C</mi></mrow><mrow is="true"><mo stretchy="false" is="true">(</mo><mi is="true">q</mi><mo is="true">,</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">δ</mi><mo is="true">,</mo><mi is="true">b</mi><mo stretchy="false" is="true">)</mo></mrow></msub></math> have dimensions k−2<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">2</mn></math> and k−3<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">−</mo><mn is="true">3</mn></math> by giving lower and upper bounds on their designed distances, which extends the results of [17]. In addition, several classes of binary and ternary self-orthogonal codes are proposed via the hulls of BCH codes and their parameters are investigated in some special cases.

External IDs:dblp:journals/ffa/LeiLWZ22