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On the Decoding of [47, 24, 11] and [48, 24, 12] Quadratic Residue Codes by Some New Fast Algorithms

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Abstract

In this paper, a new decoding algorithm, called modified reduced lookup table decoding (MRLTD) algorithm, is proposed for decoding [47, 24, 11] quadratic residue (QR) code which needs lower memory requirement compared with the algorithm based on the full lookup table and the cyclic weight decoding algorithm (CWDA). Although the memory requirements (space complexity) of the proposed algorithm are the same with a newly optimized cyclic weight (OCW) decoding algorithm, the proposed MRLTD algorithm is faster, i.e. the running-time complexity of the proposed algorithm is lower than OCW algorithm. The idea behind this decoding technique is based on the existence of a one-to-one relation between the syndromes and correctable error patterns, reported in a lookup table containing 300 syndromes which are searched by a binary search algorithm. Moreover, by a bit modification on MRLTD algorithm, it can be applied to decode effectively the extended [47, 24, 11] QR code, i.e. [48, 24, 12] QR code, applicable for correcting five errors and detecting six errors.

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References

  • Axelson J (1994) The microcontroller idea book: circuits, programs and applications featuring the 8052-basic single-chip computer, Lakeview Research

  • Chang Y, Truong TK, Reed IS, Cheng HY, Lee CD (2003) Algebraic decoding of [71, 36, 11], [79, 40, 15], and [97, 49, 15] quadratic residue codes. IEEE Trans Commun 51:1463–1473

    Article  Google Scholar 

  • Chen X, Reed IS, Truong TK (1994) Decoding the [73, 37, 13] quadratic residue code. Proc IEE 141:253–258

    Article  Google Scholar 

  • Chen YH, Truong TK, Chang Y, Lee CD, Chen SH (2007) Algebraic decoding of quadratic residue codes using Berlekamp–Massey algorithm. J Inf Sci Eng 23:127–145

    MathSciNet  Google Scholar 

  • Chen YH, Troung TK, Chien CH (2009) A lookup table decoding of systematic [47, 24, 11] quadratic residue code. Inf Sci 179:2470–2477

    Article  MathSciNet  Google Scholar 

  • Elia M (1987) Algebraic decoding of the [23, 12, 7] Golay codes. IEEE Trans Inf Theory 33:150–151

    Article  MathSciNet  Google Scholar 

  • He R, Reed IS, Truong TK, Chen X (2001) Decoding the [47, 24, 11] quadratic residue code. IEEE Trans Inf Theory 47:1181–1186

    Article  MathSciNet  Google Scholar 

  • Huang J, Zhou T, Chang HC, Xie D (2018) An optimized cyclic weight algorithm of [47, 24, 11] QR code and hardware implementation. IEEE Access 6:36995–37002

    Article  Google Scholar 

  • Lee HP, Chang HC (2014) Decoding of the triple-error-correcting binary quadratic residue codes. Inf Sci 2:7–12

    Google Scholar 

  • Lee HP, Chang HC, Lin TC, Truong TK (2008a) A weight method of decoding the [23, 12, 7] Golay code using reduced table lookup. In: International conference on communication, circuits and systems, ICCCAS, pp 1–5

  • Lee HP, Chang HC, Lin TC, Truong TK (2008b) A weight method of decoding the binary BCH code. In: Eighth international conference on intelligent systems design and applications, ISDA ’08, pp 545–549

  • Lin TC, Truong TK, Lee HP, Chang HC (2009) Algebraic decoding of the [41, 21, 9] quadratic residue code. Inf Sci 179:3451–3459

    Article  MathSciNet  Google Scholar 

  • Lin TC, Chu SI, Chang HC, Lee HP (2009) Decoding the [31, 16, 7] quadratic residue code in \(GF[2^5]\). In: 4th international conference on computer science & education, ICCSE ’09, pp 239–243

  • Lin TC, Lee HP, Chang HC, Chu SI, Truong TK (2010) High speed decoding of the binary [47, 24, 11] quadratic residue code. Inf Sci 180:4060–4068

    Article  MathSciNet  Google Scholar 

  • Lin TC, Lee HP, Chang HC, Truong TK (2012) Cyclic weight algorithm of decoding the \([47, 24, 11]\) quadratic residue code. Inf Sci 197:215–222

    Article  Google Scholar 

  • Prange E (1958) Some cyclic error-correcting codes with simple decoding algorithms. Air Force Cambridge Research Center-TN-58-156, Cambridge

    Google Scholar 

  • Reed IS, Truong TK, Chen X, Yin X (1986) The algebraic decoding of the [41, 21, 9] quadratic residue code. IEEE Trans Inf Theory 32:561–567

    Article  Google Scholar 

  • Reed IS, Yin X, Truong TK (1990a) Algebraic decoding of the [32, 16, 8] quadratic residue code. IEEE Trans Inf Theory 36:876–880

    Article  MathSciNet  Google Scholar 

  • Reed IS, Yin X, Truong TK, Holmes JK (1990b) Decoding the [24, 12, 8] Golay code. IEE Proc Comput Digit Tech 137:202–206

    Article  Google Scholar 

  • Truong TK, Chang Y, Chen YH, Lee CD (2005) Algebraic decoding of [103, 52, 19] and [113, 57, 15] quadratic residue codes. IEEE Trans Commun 53:749–754

    Article  Google Scholar 

  • Truong TK, Shih PY, Su WK, Lee CD, Chang Y (2008) Algebraic decoding of the [89, 45, 17] quadratic residue code. IEEE Trans Inf Theory 54:5005–5011

    Article  MathSciNet  Google Scholar 

  • Wicker SB (1995) Error control systems for digital communication and storage. Prentice Hall, Upper Saddle River

    MATH  Google Scholar 

  • Williams Jr LF (1976) A modification to the half-interval search (binary search) method. In: Proceedings of the 14th annual Southeast regional conference, pp 95–101

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Authors and Affiliations

  1. Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord, Iran

    Mohammad Gholami

  2. Shahrekord University, Shahrekord, Iran

    Mohammad Gholami

  3. Department of Mathematics, Shahrekord University, 115, Shahrekord, Iran

    Zeinab Roostaie

Authors
  1. Mohammad Gholami
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  2. Zeinab Roostaie
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Correspondence to Zeinab Roostaie.

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Gholami, M., Roostaie, Z. On the Decoding of [47, 24, 11] and [48, 24, 12] Quadratic Residue Codes by Some New Fast Algorithms. Iran J Sci Technol Trans Sci 45, 683–694 (2021). https://doi.org/10.1007/s40995-021-01066-8

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  • DOI: https://doi.org/10.1007/s40995-021-01066-8

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