Class of repeated‐single‐root cyclic codes for power‐line communications

Rocha, Valdemar C. da

doi:10.1049/el.2016.1877

Cited by 2 publications

(11 citation statements)

References 9 publications

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“…The two constructions presented in this Letter can be seen as generalisations of the construction presented in [7], which in both cases results for m = 1. The results in Table 1 are comparable with those in [8] with the advantage of simple implementation (see Corollary 1) of encoding and decoding operations due to the inherited cyclic code structure.…”

Section: Discussionmentioning

confidence: 55%

“…By eliminating

false(x - 1 false) g false(x false)

and its multiples from the codebook we are left with

M_{1} = p^{2} - p = p false(p - 1 false)

UCC codewords, each one consisting of a concatenation of m identical blocks of length p , i.e. having weight

m false(p - 1 false)

, where each block is a codeword of the

false(p, 2, p - 1 false)

p ‐ary cyclic code [7]. □ Theorem 2 Let

m false(x false) \neq b false(x - 1 false)

to be any non‐zero p ‐ary message polynomial of degree at most 1 in Construction 1, where

b \in GF false(p false), b \neq 0

.…”

Section: On Codeword Weights and Cyclic Ordermentioning

confidence: 99%

“…Equivalently, by multiplying (4) by

x^{p}

and reducing modulo

x^{m p} - 1

we obtain

x^{p} g false(x false) m false(x false) = g false(x false) m false(x false) mod x^{m p} - 1

.□ Theorem 3 Let

m false(x false) = x - 1

be a message polynomial in Construction 1. The resulting non‐zero codeword has cyclic order 1 and weight mp . Proof It follows from the hypothesis that

m false(x false) g false(x false) = false(x - 1 false)^{p - 1} false({false\sum}_{i = 0}^{m - 1} x^{i} false)^{p}

or, equivalently, after reduction modulo p

m false(x false) g false(x false) = ()(, \sum_{j = 0}^{p - 1} x^{j}) ()(, \sum_{i = 0}^{m - 1} x^{p i}) = \sum_{i = 0}^{m p - 1} x^{i},

where [7, Lemma 4] has been employed in the first equality. It follows from (7) that

x m false(x false) g false(x false) = m false(x false) g false(x false) mod x^{m p} - 1

, i.e.…”

Section: On Codeword Weights and Cyclic Ordermentioning

confidence: 99%

“…is precisely a codeword in the (p, 2, p − 1) p-ary cyclic code generated by the polynomial (x − 1) p−2 [7]. From 3…”

mentioning

confidence: 99%

“…has cyclic order 1 and weight mp. The non-zero scalar multiples of m(x) applied in(7) generate all p − 1 distinct codewords of cyclic order one and weight mp.□ Let m be a positive integer and let p be a prime. We consider again the factorisation ofx mp − 1 over GF(p) indicated in (1).…”

mentioning

confidence: 99%

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Uniform constant composition codes derived from repeated‐root cyclic codes

Rocha

Lemos-Neto

Alcoforado³

2018

Electron. lett.

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Uniform constant composition (UCC) codes are introduced, which are p-ary constant composition codes with block length mp where each p-ary symbol appears exactly m times in each codeword, being m a positive integer and p a prime. UCC codes are derived from a new class of p-ary repeated-root cyclic codes and two constructions are presented. Both constructions are at least asymptotically optimal with increasing p when compared with known best bounds for constant composition codes. Introduction: A constant composition code [1] is a block code defined over an alphabet with q symbols, denoted as 1, 2,. .. , q, the codebook of which consists of codewords where the symbol i occurs w i times, 1 ≤ i ≤ q, and is usually a non-linear code. We define a uniform constant composition (UCC) code as a constant composition code of block length mq, where m is a positive integer, for which w i = m, 1 ≤ i ≤ q. In this Letter, we consider only p-ary UCC codes of block length mp, where p is a prime, m is a positive integer, m ò 0 mod p, i.e. m is not a multiple of p, since the codes resulting when m ; 0 mod p represent a straightforward generalisation. Constant composition codes have recently been considered for important applications requiring non-binary alphabets [2] as, for example, power-line communications [3], construction of DNA codes [4], multiple-access communications and frequency hopping [5]. In order to construct UCC codes, we first introduce a class of p-ary (n, k, d) repeated-root cyclic codes with block length n = mp, k information symbols and minimum distance d. Two code constructions are presented, each one characterised by the corresponding p-ary generator polynomial g(x), which is a factor of x n − 1, with the remark that n = mp is the smallest value of n for which g(x) divides x n − 1. Construction 1 produces (mp, 2, m(p − 1)) codes which have a subset of M 1 = p(p − 1) UCC codewords, while Construction 2 produces (mp, m + 1, d) codes, with d = p − 1 for m = 1 and d = p for m ≥ 2, which have a subset of M 2 = p m (p − 1) UCC codewords.

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Section: Discussionmentioning

confidence: 55%

“…By eliminating

false(x - 1 false) g false(x false)

and its multiples from the codebook we are left with

M_{1} = p^{2} - p = p false(p - 1 false)

UCC codewords, each one consisting of a concatenation of m identical blocks of length p , i.e. having weight

m false(p - 1 false)

, where each block is a codeword of the

false(p, 2, p - 1 false)

p ‐ary cyclic code [7]. □ Theorem 2 Let

m false(x false) \neq b false(x - 1 false)

to be any non‐zero p ‐ary message polynomial of degree at most 1 in Construction 1, where

b \in GF false(p false), b \neq 0

.…”

Section: On Codeword Weights and Cyclic Ordermentioning

confidence: 99%

“…Equivalently, by multiplying (4) by

x^{p}

and reducing modulo

x^{m p} - 1

we obtain

x^{p} g false(x false) m false(x false) = g false(x false) m false(x false) mod x^{m p} - 1

.□ Theorem 3 Let

m false(x false) = x - 1

be a message polynomial in Construction 1. The resulting non‐zero codeword has cyclic order 1 and weight mp . Proof It follows from the hypothesis that

m false(x false) g false(x false) = false(x - 1 false)^{p - 1} false({false\sum}_{i = 0}^{m - 1} x^{i} false)^{p}

or, equivalently, after reduction modulo p

m false(x false) g false(x false) = ()(, \sum_{j = 0}^{p - 1} x^{j}) ()(, \sum_{i = 0}^{m - 1} x^{p i}) = \sum_{i = 0}^{m p - 1} x^{i},

where [7, Lemma 4] has been employed in the first equality. It follows from (7) that

x m false(x false) g false(x false) = m false(x false) g false(x false) mod x^{m p} - 1

, i.e.…”

Section: On Codeword Weights and Cyclic Ordermentioning

confidence: 99%

“…is precisely a codeword in the (p, 2, p − 1) p-ary cyclic code generated by the polynomial (x − 1) p−2 [7]. From 3…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Uniform constant composition codes derived from repeated‐root cyclic codes

Rocha

Lemos-Neto

Alcoforado³

2018

Electron. lett.

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The Construction of Permutation Group Codes for Communication Systems: Prime or Prime Power?

2020

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This paper focuses on correcting a theorem relating to the construction of permutation group codes (PGCs). The theorem in question assumed that affine transformation could be used to enumerate all the code words of a permutation array with a minimum Hamming distance (MHD) of n − 1 for any n > 1. This assumption was founded upon the proposition that, if the code length, n, is a prime power, then the maximum cardinality of the code will be n(n − 1) and its MHD will be n − 1. However, two typical algebraic methods, one relying on affine transformation and another upon the composite operation of two small subgroups of a symmetric group, can violate this proposition. This is because it is only when n is a prime rather than a prime power that they can enumerate all the code words of an (n; n (n − 1) ; n − 1)-PGC. By investigating how the range of n impacts upon the cardinality and MHD of a code, we provide a corrective theorem that stipulates the construction of (p q ; p 2q (1 − 1/p); p q (1 − 1/p))-PGCs when n = p q is a prime power, where p is a prime and q > 1. On the basis of this theorem, and under the condition of n being the power of 2, we construct a (2 q ; 2 2q−1 ; 2 q−1 )-PGC and present an encoder that can map a k-bit binary information sequence to an n = 2 2q -dimension permutation code word. The natural array structure of (2 q ; 2 2q−1 ; 2 q−1 )-PGCs makes them especially well-suited to forming the basis of low-complexity encoders. We present simulation-based experiments that show that, as the code length increases, the performance of these codes improves. The best performance is for a (16; 128; 8)-PGC, which can achieve −3.8 dB with a word error rate of 10 −7 . INDEX TERMS Channel coding, error correcting code, permutation array, permutation group code, mapping encoder.

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Class of repeated‐single‐root cyclic codes for power‐line communications

Cited by 2 publications

References 9 publications

Uniform constant composition codes derived from repeated‐root cyclic codes

Uniform constant composition codes derived from repeated‐root cyclic codes

The Construction of Permutation Group Codes for Communication Systems: Prime or Prime Power?

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