Abstract
We introduce a consistent and efficient method to construct self-dual 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">$GF(q)$ </tex-math></inline-formula> using symmetric matrices and eigenvectors from a self-dual code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$GF(q)$ </tex-math></inline-formula> of smaller length where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q \equiv 1 \pmod 4$ </tex-math></inline-formula> . Using this method, which is called a ‘symmetric building-up’ construction, we improve the bounds of the best-known minimum weights of self-dual codes with lengths up to 40, which have not significantly improved for almost two decades. We focus on a class of self-dual codes, which includes double circulant codes. We obtain 2967 new self-dual 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">$GF(13)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$GF(17)$ </tex-math></inline-formula> up to equivalence. We also compute the minimum weights of quadratic residue(QR) codes that were previously unknown. These are a [20, 10, 10] QR self-dual code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$GF(23)$ </tex-math></inline-formula> , [24, 12, 12] QR self-dual 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">$GF(29)$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$GF(41)$ </tex-math></inline-formula> , and a [32, 16, 14] QR self-dual code over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$GF(19)$ </tex-math></inline-formula> . They have the highest minimum weights so far.
Highlights
The theory of error-correcting code, which was born with the invention of computers, has been an interesting topic of mathematics as well as industry, such as satellites, CD players, and cellular phones
We show that every symmetric self-dual code of length 2n + 2 is constructed from a symmetric self-dual code of length 2n up to equivalence by using this construction method
We remark that many maximal distance separable (MDS) and optimal self-dual codes are obtained by using the construction method of pure double circulant codes and bordered double circulant codes in [3], [16]
Summary
The theory of error-correcting code, which was born with the invention of computers, has been an interesting topic of mathematics as well as industry, such as satellites, CD players, and cellular phones. We present all of the up-to-date results concerning minimum weight bounds and the existence of optimal self-dual codes in Tables 1, 2, and 3. We try to improve the bounds of minimum weights by constructing self-dual codes of long length as many as possible. To this end, we investigate the consistent and efficient method to construct self-dual codes. Since we obtain [18,9,8] self-dual codes over GF (13), the bound of the highest minimum distance of selfdual code over GF (13) of length 18 is turned to 8-9. All computations in this paper were done with the computer algebra system Magma [6]
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