Binary cyclic-gap constant weight codes with low-complexity encoding and decoding

In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size M = 2 k so that codewords can conveniently be labeled with binary vectors of length k . For every integer ℓ ≥ 3 , we construct a ( n = 2 ℓ , M = 2 k ℓ...

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Vydáno v:	Designs, codes, and cryptography Ročník 92; číslo 12; s. 4247 - 4277
Hlavní autoři:	Sasidharan, Birenjith, Viterbo, Emanuele, Dau, Son Hoang
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.12.2024 Springer Nature B.V
Témata:	Algorithms Binomial coefficients Codes Coding Coding and Information Theory Complexity Computer Science Construction Cryptology Decoding Discrete Mathematics in Computer Science Integers Parameter modification Upper bounds Enumerative coding 94B15 Nonlinear codes 94B25 94B35 Cyclic-gap code 94A29 Constant weight codes Binary codes 68P30 Low complexity
ISSN:	0925-1022, 1573-7586
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Abstract	In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size M = 2 k so that codewords can conveniently be labeled with binary vectors of length k . For every integer ℓ ≥ 3 , we construct a ( n = 2 ℓ , M = 2 k ℓ , d = 2 ) constant weight code C [ ℓ ] of weight ℓ by encoding information in the gaps between successive 1’s of a vector, and call them as cyclic-gap constant weight codes. The code is associated with a finite integer sequence of length ℓ satisfying a constraint defined as anchor-decodability that is pivotal to ensure low complexity for encoding and decoding. The time complexity of the encoding algorithm is linear in the input size k , and that of the decoding algorithm is poly-logarithmic in the input size n , discounting the linear time spent on parsing the input. Both the algorithms do not require expensive computation of binomial coefficients, unlike the case in many existing schemes. Among codes generated by all anchor-decodable sequences, we show that C [ ℓ ] has the maximum size with k ℓ ≥ ℓ 2 - ℓ log 2 ℓ + log 2 ℓ - 0.279 ℓ - 0.721 . As k is upper bounded by ℓ 2 - ℓ log 2 ℓ + O ( ℓ ) information-theoretically, the code C [ ℓ ] is optimal in its size with respect to two higher order terms of ℓ . In particular, k ℓ meets the upper bound for ℓ = 3 and one-bit away for ℓ = 4 . On the other hand, we show that C [ ℓ ] is not unique in attaining k ℓ by constructing an alternate code C ^ [ ℓ ] again parameterized by an integer ℓ ≥ 3 with a different low-complexity decoder, yet having the same size 2 k ℓ when 3 ≤ ℓ ≤ 7 . Finally, we also derive new codes by modifying C [ ℓ ] that offer a wider range on blocklength and weight while retaining low complexity for encoding and decoding. For certain selected values of parameters, these modified codes too have an optimal k .
AbstractList	In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size M = 2 k so that codewords can conveniently be labeled with binary vectors of length k . For every integer ℓ ≥ 3 , we construct a ( n = 2 ℓ , M = 2 k ℓ , d = 2 ) constant weight code C [ ℓ ] of weight ℓ by encoding information in the gaps between successive 1’s of a vector, and call them as cyclic-gap constant weight codes. The code is associated with a finite integer sequence of length ℓ satisfying a constraint defined as anchor-decodability that is pivotal to ensure low complexity for encoding and decoding. The time complexity of the encoding algorithm is linear in the input size k , and that of the decoding algorithm is poly-logarithmic in the input size n , discounting the linear time spent on parsing the input. Both the algorithms do not require expensive computation of binomial coefficients, unlike the case in many existing schemes. Among codes generated by all anchor-decodable sequences, we show that C [ ℓ ] has the maximum size with k ℓ ≥ ℓ 2 - ℓ log 2 ℓ + log 2 ℓ - 0.279 ℓ - 0.721 . As k is upper bounded by ℓ 2 - ℓ log 2 ℓ + O ( ℓ ) information-theoretically, the code C [ ℓ ] is optimal in its size with respect to two higher order terms of ℓ . In particular, k ℓ meets the upper bound for ℓ = 3 and one-bit away for ℓ = 4 . On the other hand, we show that C [ ℓ ] is not unique in attaining k ℓ by constructing an alternate code C ^ [ ℓ ] again parameterized by an integer ℓ ≥ 3 with a different low-complexity decoder, yet having the same size 2 k ℓ when 3 ≤ ℓ ≤ 7 . Finally, we also derive new codes by modifying C [ ℓ ] that offer a wider range on blocklength and weight while retaining low complexity for encoding and decoding. For certain selected values of parameters, these modified codes too have an optimal k . In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size $$M=2^k$$ M = 2 k so that codewords can conveniently be labeled with binary vectors of length k . For every integer $$\ell \ge 3$$ ℓ ≥ 3 , we construct a $$(n=2^\ell , M=2^{k_{\ell }}, d=2)$$ ( n = 2 ℓ , M = 2 k ℓ , d = 2 ) constant weight code $${{{\mathcal {C}}}[\ell ]$$ C [ ℓ ] of weight $$\ell $$ ℓ by encoding information in the gaps between successive 1’s of a vector, and call them as cyclic-gap constant weight codes. The code is associated with a finite integer sequence of length $$\ell $$ ℓ satisfying a constraint defined as anchor-decodability that is pivotal to ensure low complexity for encoding and decoding. The time complexity of the encoding algorithm is linear in the input size k , and that of the decoding algorithm is poly-logarithmic in the input size n , discounting the linear time spent on parsing the input. Both the algorithms do not require expensive computation of binomial coefficients, unlike the case in many existing schemes. Among codes generated by all anchor-decodable sequences, we show that $${{{\mathcal {C}}}[\ell ]$$ C [ ℓ ] has the maximum size with $$k_{\ell } \ge \ell ^2-\ell \log _2\ell + \log _2\ell - 0.279\ell - 0.721$$ k ℓ ≥ ℓ 2 - ℓ log 2 ℓ + log 2 ℓ - 0.279 ℓ - 0.721 . As k is upper bounded by $$\ell ^2-\ell \log _2\ell +O(\ell )$$ ℓ 2 - ℓ log 2 ℓ + O ( ℓ ) information-theoretically, the code $${{{\mathcal {C}}}[\ell ]$$ C [ ℓ ] is optimal in its size with respect to two higher order terms of $$\ell $$ ℓ . In particular, $$k_\ell $$ k ℓ meets the upper bound for $$\ell =3$$ ℓ = 3 and one-bit away for $$\ell =4$$ ℓ = 4 . On the other hand, we show that $${{{\mathcal {C}}}[\ell ]$$ C [ ℓ ] is not unique in attaining $$k_{\ell }$$ k ℓ by constructing an alternate code $$\mathcal{{\hat{C}}}[\ell ]$$ C ^ [ ℓ ] again parameterized by an integer $$\ell \ge 3$$ ℓ ≥ 3 with a different low-complexity decoder, yet having the same size $$2^{k_{\ell }}$$ 2 k ℓ when $$3 \le \ell \le 7$$ 3 ≤ ℓ ≤ 7 . Finally, we also derive new codes by modifying $${{{\mathcal {C}}}[\ell ]$$ C [ ℓ ] that offer a wider range on blocklength and weight while retaining low complexity for encoding and decoding. For certain selected values of parameters, these modified codes too have an optimal k . In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size M=2k so that codewords can conveniently be labeled with binary vectors of length k. For every integer ℓ≥3, we construct a (n=2ℓ,M=2kℓ,d=2) constant weight code C[ℓ] of weight ℓ by encoding information in the gaps between successive 1’s of a vector, and call them as cyclic-gap constant weight codes. The code is associated with a finite integer sequence of length ℓ satisfying a constraint defined as anchor-decodability that is pivotal to ensure low complexity for encoding and decoding. The time complexity of the encoding algorithm is linear in the input size k, and that of the decoding algorithm is poly-logarithmic in the input size n, discounting the linear time spent on parsing the input. Both the algorithms do not require expensive computation of binomial coefficients, unlike the case in many existing schemes. Among codes generated by all anchor-decodable sequences, we show that C[ℓ] has the maximum size with kℓ≥ℓ2-ℓlog2ℓ+log2ℓ-0.279ℓ-0.721. As k is upper bounded by ℓ2-ℓlog2ℓ+O(ℓ) information-theoretically, the code C[ℓ] is optimal in its size with respect to two higher order terms of ℓ. In particular, kℓ meets the upper bound for ℓ=3 and one-bit away for ℓ=4. On the other hand, we show that C[ℓ] is not unique in attaining kℓ by constructing an alternate code C^[ℓ] again parameterized by an integer ℓ≥3 with a different low-complexity decoder, yet having the same size 2kℓ when 3≤ℓ≤7. Finally, we also derive new codes by modifying C[ℓ] that offer a wider range on blocklength and weight while retaining low complexity for encoding and decoding. For certain selected values of parameters, these modified codes too have an optimal k.
Author	Viterbo, Emanuele Dau, Son Hoang Sasidharan, Birenjith
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Cites_doi	10.1109/TIT.1980.1056141 10.1109/18.370146 10.1109/18.887851 10.1109/TIT.2011.2145490 10.1017/CBO9780511813498 10.1109/PROC.1965.3680 10.1109/18.370117 10.1109/TIT.2009.2021366 10.3390/a15020036 10.1016/j.disc.2007.11.048 10.3390/a14030097 10.1145/360767.360811 10.1109/18.59932 10.1109/TIT.1972.1054832 10.1109/TIT.1962.1057714 10.1109/18.53748 10.1080/00207168508803468 10.1016/S0924-6509(01)80051-9 10.1109/TIT.1973.1054929 10.1090/psapm/010/0113289 10.1109/ICTEL.2003.1191603 10.1109/ISIT.2005.1523371 10.1109/DCC.2003.1194039
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Snippet	In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size M = 2 k so... In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size $$M=2^k$$... In this paper, we focus on the design of binary constant weight codes that admit low-complexity encoding and decoding algorithms, and that have size M=2k so...
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SubjectTerms	Algorithms Binomial coefficients Codes Coding Coding and Information Theory Complexity Computer Science Construction Cryptology Decoding Discrete Mathematics in Computer Science Integers Parameter modification Upper bounds
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Title	Binary cyclic-gap constant weight codes with low-complexity encoding and decoding
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