{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T20:16:55Z","timestamp":1777666615260,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,8]],"date-time":"2020-01-08T00:00:00Z","timestamp":1578441600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Research Grants - ARUT of the TUIASI","award":["GnaC2018_39"],"award-info":[{"award-number":["GnaC2018_39"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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     16 \u03a8     or     48 \u03a8    , where    \u03a8    is a product of different prime numbers greater than three. Some coefficient restrictions are applied when for a prime      p i  \u2223 \u03a8    , condition     3 \u2224 (  p i  \u2212 1 )     is fulfilled. Two upper bounds are obtained for different classes of 4-PP coefficients. For a 4-PP      f 4   x 4  +  f 3   x 3  +  f 2   x 2  +  f 1  x   ( mod  16  k L  \u03a8 )     ,      k L  \u2208  { 1 , 3 }     , the upper bound of 28 is obtained when the coefficient     f 3     of the equivalent 4-permutation polynomials (PPs) fulfills      f 3  \u2208  { 0 , 4 \u03a8 }      or when      f 3  \u2208  { 2 \u03a8 , 6 \u03a8 }      and      f 2  \u2208  {  ( 4  k L  \u2212 1 )  \u00b7 \u03a8 ,  ( 8  k L  \u2212 1 )  \u00b7 \u03a8 }     ,      k L  \u2208  { 1 , 3 }     , for any values of the other coefficients. The upper bound of 36 is obtained when the coefficient     f 3     of the equivalent 4-PPs fulfills      f 3  \u2208  { 2 \u03a8 , 6 \u03a8 }      and      f 2  \u2208  {  ( 2  k L  \u2212 1 )  \u00b7 \u03a8 ,  ( 6  k L  \u2212 1 )  \u00b7 \u03a8 }     ,      k L  \u2208  { 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      f 4  ,  f 3      and     f 2    . 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.<\/jats:p>","DOI":"10.3390\/e22010078","type":"journal-article","created":{"date-parts":[[2020,1,9]],"date-time":"2020-01-09T03:07:11Z","timestamp":1578539231000},"page":"78","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Lengths for Which Fourth Degree PP Interleavers Lead to Weaker Performances Compared to Quadratic and Cubic PP Interleavers"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1697-6406","authenticated-orcid":false,"given":"Lucian","family":"Trifina","sequence":"first","affiliation":[{"name":"Department of Telecommunications and Information Technologies, \u201cGheorghe Asachi\u201d Technical University, 700506 Iasi, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2003-6644","authenticated-orcid":false,"given":"Daniela","family":"Tarniceriu","sequence":"additional","affiliation":[{"name":"Department of Telecommunications and Information Technologies, \u201cGheorghe Asachi\u201d Technical University, 700506 Iasi, Romania"}]},{"given":"Jonghoon","family":"Ryu","sequence":"additional","affiliation":[{"name":"Samsung Electronics, Inc., Suwon 16677, Korea"}]},{"given":"Ana-Mirela","family":"Rotopanescu","sequence":"additional","affiliation":[{"name":"Department of Telecommunications and Information Technologies, \u201cGheorghe Asachi\u201d Technical University, 700506 Iasi, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1109\/COMST.2019.2893851","article-title":"Survey of turbo, LDPC, and polar decoder ASIC implementations","volume":"21","author":"Shao","year":"2019","journal-title":"IEEE Commun. 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