New Bounds on the Size of Binary Codes With Large Minimum Distance
Let <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> denote the maximum size of a binary code of length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and minimum Hamming d...
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| Veröffentlicht in: | IEEE journal on selected areas in information theory Jg. 4; S. 219 - 231 |
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| Sprache: | Englisch |
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2023
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| Abstract | Let <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> denote the maximum size of a binary code of length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and minimum Hamming distance <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>. Studying <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula>, including efforts to determine it as well to derive bounds on <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> for large <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>'s, is one of the most fundamental subjects in coding theory. In this paper, we explore new lower and upper bounds on <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> in the large-minimum distance regime, in particular, when <inline-formula> <tex-math notation="LaTeX">d = n/2 - \Omega (\sqrt {n}) </tex-math></inline-formula>. We first provide a new construction of cyclic codes, by carefully selecting specific roots in the binary extension field for the check polynomial, with length <inline-formula> <tex-math notation="LaTeX">n= 2^{m} -1 </tex-math></inline-formula>, distance <inline-formula> <tex-math notation="LaTeX">d \geq n/2 - 2^{c-1}\sqrt {n} </tex-math></inline-formula>, and size <inline-formula> <tex-math notation="LaTeX">n^{c+1/2} </tex-math></inline-formula>, for any <inline-formula> <tex-math notation="LaTeX">m\geq 4 </tex-math></inline-formula> and any integer <inline-formula> <tex-math notation="LaTeX">c </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">0 \leq c \leq m/2 - 1 </tex-math></inline-formula>. These code parameters are slightly worse than those of the Delsarte-Goethals (DG) codes that provide the previously known best lower bound in the large-minimum distance regime. However, using a similar and extended code construction technique we show a sequence of cyclic codes that improve upon DG codes and provide the best lower bound in a narrower range of the minimum distance <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>, in particular, when <inline-formula> <tex-math notation="LaTeX">d = n/2 - \Omega (n^{2/3}) </tex-math></inline-formula>. Furthermore, by leveraging a Fourier-analytic view of Delsarte's linear program, upper bounds on <inline-formula> <tex-math notation="LaTeX">A(n, \left \lceil{ n/2 - \rho \sqrt {n}\, }\right \rceil) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\rho \in (0.5, 9.5) </tex-math></inline-formula> are obtained that scale polynomially in <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>. To the best of authors' knowledge, the upper bound due to Barg and Nogin (2006) is the only previously known upper bound that scale polynomially in <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> in this regime. We numerically demonstrate that our upper bound improves upon the Barg-Nogin upper bound in the specified high-minimum distance regime. |
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| AbstractList | Let [Formula Omitted] denote the maximum size of a binary code of length [Formula Omitted] and minimum Hamming distance [Formula Omitted]. Studying [Formula Omitted], including efforts to determine it as well to derive bounds on [Formula Omitted] for large [Formula Omitted]’s, is one of the most fundamental subjects in coding theory. In this paper, we explore new lower and upper bounds on [Formula Omitted] in the large-minimum distance regime, in particular, when [Formula Omitted]. We first provide a new construction of cyclic codes, by carefully selecting specific roots in the binary extension field for the check polynomial, with length [Formula Omitted], distance [Formula Omitted], and size [Formula Omitted], for any [Formula Omitted] and any integer [Formula Omitted] with [Formula Omitted]. These code parameters are slightly worse than those of the Delsarte–Goethals (DG) codes that provide the previously known best lower bound in the large-minimum distance regime. However, using a similar and extended code construction technique we show a sequence of cyclic codes that improve upon DG codes and provide the best lower bound in a narrower range of the minimum distance [Formula Omitted], in particular, when [Formula Omitted]. Furthermore, by leveraging a Fourier-analytic view of Delsarte’s linear program, upper bounds on [Formula Omitted] with [Formula Omitted] are obtained that scale polynomially in [Formula Omitted]. To the best of authors’ knowledge, the upper bound due to Barg and Nogin (2006) is the only previously known upper bound that scale polynomially in [Formula Omitted] in this regime. We numerically demonstrate that our upper bound improves upon the Barg-Nogin upper bound in the specified high-minimum distance regime. Let <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> denote the maximum size of a binary code of length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> and minimum Hamming distance <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>. Studying <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula>, including efforts to determine it as well to derive bounds on <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> for large <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>'s, is one of the most fundamental subjects in coding theory. In this paper, we explore new lower and upper bounds on <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> in the large-minimum distance regime, in particular, when <inline-formula> <tex-math notation="LaTeX">d = n/2 - \Omega (\sqrt {n}) </tex-math></inline-formula>. We first provide a new construction of cyclic codes, by carefully selecting specific roots in the binary extension field for the check polynomial, with length <inline-formula> <tex-math notation="LaTeX">n= 2^{m} -1 </tex-math></inline-formula>, distance <inline-formula> <tex-math notation="LaTeX">d \geq n/2 - 2^{c-1}\sqrt {n} </tex-math></inline-formula>, and size <inline-formula> <tex-math notation="LaTeX">n^{c+1/2} </tex-math></inline-formula>, for any <inline-formula> <tex-math notation="LaTeX">m\geq 4 </tex-math></inline-formula> and any integer <inline-formula> <tex-math notation="LaTeX">c </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">0 \leq c \leq m/2 - 1 </tex-math></inline-formula>. These code parameters are slightly worse than those of the Delsarte-Goethals (DG) codes that provide the previously known best lower bound in the large-minimum distance regime. However, using a similar and extended code construction technique we show a sequence of cyclic codes that improve upon DG codes and provide the best lower bound in a narrower range of the minimum distance <inline-formula> <tex-math notation="LaTeX">d </tex-math></inline-formula>, in particular, when <inline-formula> <tex-math notation="LaTeX">d = n/2 - \Omega (n^{2/3}) </tex-math></inline-formula>. Furthermore, by leveraging a Fourier-analytic view of Delsarte's linear program, upper bounds on <inline-formula> <tex-math notation="LaTeX">A(n, \left \lceil{ n/2 - \rho \sqrt {n}\, }\right \rceil) </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">\rho \in (0.5, 9.5) </tex-math></inline-formula> are obtained that scale polynomially in <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>. To the best of authors' knowledge, the upper bound due to Barg and Nogin (2006) is the only previously known upper bound that scale polynomially in <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> in this regime. We numerically demonstrate that our upper bound improves upon the Barg-Nogin upper bound in the specified high-minimum distance regime. |
| Author | Pang, James Chin-Jen Pradhan, S. Sandeep Mahdavifar, Hessam |
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| Snippet | Let <inline-formula> <tex-math notation="LaTeX">A(n, d) </tex-math></inline-formula> denote the maximum size of a binary code of length <inline-formula>... Let [Formula Omitted] denote the maximum size of a binary code of length [Formula Omitted] and minimum Hamming distance [Formula Omitted]. Studying [Formula... |
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| SubjectTerms | Binary codes Codes Eigenvalues and eigenfunctions Error correction codes Hamming distances Harmonic analysis Information theory Lower bounds Polynomials Reed-Muller codes Upper bound Upper bounds |
| Title | New Bounds on the Size of Binary Codes With Large Minimum Distance |
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