Trifina L, Tarniceriu D, Ryu J, Rotopanescu AM. Lengths for Which Fourth Degree PP Interleavers Lead to Weaker Performances Compared to Quadratic and Cubic PP Interleavers.
Entropy (Basel) 2020;
22:e22010078. [PMID:
33285853 PMCID:
PMC7516510 DOI:
10.3390/e22010078]
[Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/25/2019] [Revised: 12/31/2019] [Accepted: 01/06/2020] [Indexed: 11/16/2022]
Abstract
In this paper, we obtain upper bounds on the minimum distance for turbo codes using fourth degree permutation polynomial (4-PP) interleavers of a specific interleaver length and classical turbo codes of nominal 1/3 coding rate, with two recursive systematic convolutional component codes with generator matrix G=[1,15/13]. The interleaver lengths are of the form 16Ψ or 48Ψ, where Ψ is a product of different prime numbers greater than three. Some coefficient restrictions are applied when for a prime pi∣Ψ, condition 3∤(pi−1) is fulfilled. Two upper bounds are obtained for different classes of 4-PP coefficients. For a 4-PP f4x4+f3x3+f2x2+f1x(mod16kLΨ), kL∈{1,3}, the upper bound of 28 is obtained when the coefficient f3 of the equivalent 4-permutation polynomials (PPs) fulfills f3∈{0,4Ψ} or when f3∈{2Ψ,6Ψ} and f2∈{(4kL−1)·Ψ,(8kL−1)·Ψ}, kL∈{1,3}, for any values of the other coefficients. The upper bound of 36 is obtained when the coefficient f3 of the equivalent 4-PPs fulfills f3∈{2Ψ,6Ψ} and f2∈{(2kL−1)·Ψ,(6kL−1)·Ψ}, kL∈{1,3}, for any values of the other coefficients. Thus, the task of finding out good 4-PP interleavers of the previous mentioned lengths is highly facilitated by this result because of the small range required for coefficients f4,f3 and f2. It was also proven, by means of nonlinearity degree, that for the considered inteleaver lengths, cubic PPs and quadratic PPs with optimum minimum distances lead to better error rate performances compared to 4-PPs with optimum minimum distances.
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