m -ary Balanced Codes With Parallel Decoding
An m-ary block code, m = 2, 3, 4,..., of length n ϵ IN is called balanced if, and only if, every codeword is balanced; that is, the real sum of the codeword components, or weight, is equal to ⌊(m - 1)n/2⌋. This paper presents efficient encoding schemes to m-ary balanced codes with parallel (hence, f...
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| Vydáno v: | IEEE transactions on information theory Ročník 61; číslo 6; s. 3251 - 3264 |
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| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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IEEE
01.06.2015
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | An m-ary block code, m = 2, 3, 4,..., of length n ϵ IN is called balanced if, and only if, every codeword is balanced; that is, the real sum of the codeword components, or weight, is equal to ⌊(m - 1)n/2⌋. This paper presents efficient encoding schemes to m-ary balanced codes with parallel (hence, fast) decoding. In fact, the decoding time complexity is O(1) digit operations. These schemes are a generalization to the m-ary alphabet of Knuth's complementation method with parallel decoding. Let ( n w ) m indicate the number of m-ary words w of length n and weight w ϵ(0,1, ... , (m - 1)n}. For any m ϵ IN, m ≥ 2, a simple implementation of the method is given which uses r ϵ IN check digits to balance k ≤ {(⌊(m-1) r /2⌋) m - (m mod 2 + [(m - 1)k] mod 2}}/(m - 1) information digits with an encoding time complexity of O(mk log m k) digit operations. A refined implementation of the parallel decoding method is also given with r check digits and k ≤ (m r -1)/(m -1) information digits, where the encoding time complexity is O(k√log m k). Thus, the proposed codes are less redundant than the m-ary balanced codes with parallel decoding found in the literature and yet maintain the same complexity. |
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| AbstractList | An m-ary block code, m = 2, 3, 4,..., of length n ϵ IN is called balanced if, and only if, every codeword is balanced; that is, the real sum of the codeword components, or weight, is equal to ⌊(m - 1)n/2⌋. This paper presents efficient encoding schemes to m-ary balanced codes with parallel (hence, fast) decoding. In fact, the decoding time complexity is O(1) digit operations. These schemes are a generalization to the m-ary alphabet of Knuth's complementation method with parallel decoding. Let ( n w ) m indicate the number of m-ary words w of length n and weight w ϵ(0,1, ... , (m - 1)n}. For any m ϵ IN, m ≥ 2, a simple implementation of the method is given which uses r ϵ IN check digits to balance k ≤ {(⌊(m-1) r /2⌋) m - (m mod 2 + [(m - 1)k] mod 2}}/(m - 1) information digits with an encoding time complexity of O(mk log m k) digit operations. A refined implementation of the parallel decoding method is also given with r check digits and k ≤ (m r -1)/(m -1) information digits, where the encoding time complexity is O(k√log m k). Thus, the proposed codes are less redundant than the m-ary balanced codes with parallel decoding found in the literature and yet maintain the same complexity. An $m$ -ary block code, $m=2,3,4,ldots $ , of length $n !in ! ...mathbf...I...!...mathbf...I...mskip -7mu...mathbf...N... $ is called balanced if, and only if, every codeword is balanced; that is, the real sum of the codeword components, or weight, is equal to $ left lfloor... (m-1)n/2 ...right rfloor $ . This paper presents efficient encoding schemes to $m$ -ary balanced codes with parallel (hence, fast) decoding. In fact, the decoding time complexity is $O(1)$ digit operations. These schemes are a generalization to the $m$ -ary alphabet of Knuth's complementation method with parallel decoding. Let $binom...n... w..._...m...$ indicate the number of $m$ -ary words of length $n$ and weight $w !in !...0,1,ldots ,(m-1)n...$ . For any $m !in ! ...mathbf...I...!...mathbf...I...mskip -7mu...mathbf...N... $ , $mgeq 2$ , a simple implementation of the method is given which uses $r !in ! ...mathbf...I...!...mathbf...I...mskip -7mu...mathbf...N... $ check digits to balance $kleq ...binom...r... ... left lfloor... (m-1)r/2 ...right rfloor ..._...vphantom ...R_...R_...m...-...mbmod ...2...+[(m-1)k]bmod 2.../(m-1)$ information digits with an encoding time complexity of $O(mklog _...m...k)$ digit operations. A refined implementation of the parallel decoding method is also given with $r$ check digits and $kleq (m...r...-1)/(m-1)$ information digits, where the encoding time complexity is $O(ksqrt ...log _...m...k...)$ . Thus, the proposed codes are less redundant than the $m$ -ary balanced codes with parallel decoding found in the literature and yet maintain the same complexity. (ProQuest: ... denotes formulae/symbols omitted.) |
| Author | Elmougy, Samir Bose, Bella Tallini, Luca G. Pelusi, Danilo |
| Author_xml | – sequence: 1 givenname: Danilo surname: Pelusi fullname: Pelusi, Danilo email: dpelusi@unite.it organization: Fac. di Sci. della Comun., Univ. degli Studi di Teramo, Teramo, Italy – sequence: 2 givenname: Samir surname: Elmougy fullname: Elmougy, Samir email: mougy@ksu.edu.sa organization: Dept. of Comput. Sci., King Saud Univ., Riyadh, Saudi Arabia – sequence: 3 givenname: Luca G. surname: Tallini fullname: Tallini, Luca G. email: ltallini@unite.it organization: Fac. di Sci. della Comun., Univ. degli Studi di Teramo, Teramo, Italy – sequence: 4 givenname: Bella surname: Bose fullname: Bose, Bella email: bose@eecs.orst.edu organization: Sch. of Electr. Eng. & Comput. Sci., Oregon State Univ., Corvallis, OR, USA |
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| Keywords | optical and magnetic recording Balanced codes parallel decoding scheme unidirectional error detection m -ary alphabet Knuth’s complementation method |
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| Snippet | An m-ary block code, m = 2, 3, 4,..., of length n ϵ IN is called balanced if, and only if, every codeword is balanced; that is, the real sum of the codeword... An $m$ -ary block code, $m=2,3,4,ldots $ , of length $n !in ! ...mathbf...I...!...mathbf...I...mskip -7mu...mathbf...N... $ is called balanced if, and only if,... |
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| SubjectTerms | Balanced codes Coding theory Combinatorics Decoding Electronics Encoding Frequency modulation Indexes Knuth's complementation method m-ary alphabet optical and magnetic recording parallel decoding scheme Redundancy Time complexity unidirectional error detection |
| Title | m -ary Balanced Codes With Parallel Decoding |
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