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Research Letters in Communications
Volumeย 2009, Article IDย 107432, 3 pages
Research Letter

Decoding the Ternary (23, 11, 9) Quadratic Residue Code

Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, USA

Received 11 January 2009; Accepted 10 April 2009

Academic Editor: Guosenย Yue

Copyright ยฉ 2009 J. Carmelo Interlando. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The algebraic decoding of binary quadratic residue codes can be performed using the Peterson or the Berlekamp-Massey algorithm once certain unknown syndromes are determined or eliminated. The technique of determining unknown syndromes is applied to the nonbinary case to decode the expurgated ternary quadratic residue code of length 23.

1. Introduction

Quadratic residue (QR) codes are cyclic, nominally half-rate codes, that are powerful with respect to their error-correction capabilities. Decoding QR codes is in general a difficult task, but great progress has been made in the binary case since the work of Elia [1] and He et al. [2]. Decoding algorithms for certain nonbinary QR codes were proposed by Higgs and Humphreys in [3] and [4]. In [5], decoding of QR codes is performed by embedding them in codes over cyclotomic number fields.

This paper shows that one technique used to decode binary QR codes can be applied successfully to decode nonbinary QR codes. The main idea is to determine certain unknown syndromes in order to restore linearity to Newton's identities. Once this is done, either the Peterson or the Berlekamp-Massey algorithm can be used to solve the identities. The method of determining unknown syndromes was first presented by He et al. in [2] to decode the binary QR code of length 47 and subsequently to decode several other binary QR codes; see [6] and references therein.

Section 2 reviews the necessary background and the latter method, with the objective of establishing notation. In Section 3, the method is illustrated on the decoding of the expurgated ternary QR code of length 23. The focus is solely on the calculation of the error-location polynomial. Error values can be found from the evaluator polynomial [7, p. 246] once the error locations are determined.

2. Background and Terminology

Let ๐’ฌ={1,2,3,4,6,8,9,12,13,16,18} be the set of quadratic residues of 23 and ๐’ฉ the set of quadratic nonresidues of 23. The smallest extension of ๐”ฝ3=GF(3) containing ๐›ผ, a primitive twenty-third root of unity, is ๐”ฝ311=GF(311). Denote the set {0}โˆช๐’ฌ by ๐’ต and define ๐‘”(๐‘ฅ)โˆˆ๐”ฝ3[๐‘ฅ] as๎‘๐‘”(๐‘ฅ)=๐‘–โˆˆ๐’ต๎€ท๐‘ฅโˆ’๐›ผ๐‘–๎€ธ=๐‘ฅ12+๐‘ฅ9+๐‘ฅ7+๐‘ฅ6+2๐‘ฅ5+๐‘ฅ4+2๐‘ฅ3+2๐‘ฅ+1.(1)The cyclic code generated by ๐‘”(๐‘ฅ) is the expurgated ternary QR of length 23; see [7]. Its minimum Hamming distance is equal to 9, which can be verified by direct inspection.

Let โˆ‘๐‘(๐‘ฅ)=22๐‘–=0๐‘๐‘–๐‘ฅ๐‘–โˆˆ๐”ฝ3[๐‘ฅ] be the sent code polynomial, that is, a multiple of ๐‘”(๐‘ฅ). The received polynomial, denoted by โˆ‘๐‘Ÿ(๐‘ฅ)=22๐‘–=0๐‘Ÿ๐‘–๐‘ฅ๐‘–, satisfies ๐‘Ÿ(๐‘ฅ)=๐‘(๐‘ฅ)+๐‘’(๐‘ฅ) where โˆ‘๐‘’(๐‘ฅ)=22๐‘–=0๐‘’๐‘–๐‘ฅ๐‘–โˆˆ๐”ฝ3[๐‘ฅ] is the error pattern. Let ๐œˆ denote the Hamming weight of ๐‘’(๐‘ฅ). Observe that ๐‘’(๐‘ฅ) can be correctly determined provided ๐œˆโ‰ค4. Only ๐‘”(๐‘ฅ) and ๐‘Ÿ(๐‘ฅ) are known to the receiver, which seeks to determine the most probable ๐‘’(๐‘ฅ). For any ๐‘˜โˆˆโ„ค, the syndrome ๐‘ ๐‘˜ is defined as ๐‘ ๐‘˜=๐‘’(๐›ผ๐‘˜). It follows that ๐‘ 3๐‘˜=๐‘ 3๐‘˜, for all ๐‘˜โˆˆโ„ค. Observe that for all ๐‘˜,โ„“โˆˆโ„ค, ๐‘ ๐‘˜=๐‘ โ„“ whenever ๐‘˜โ‰กโ„“(mod23). For any ๐‘˜โˆˆ๐’ต,๐‘”(๐›ผ๐‘˜)=0, whence ๐‘ ๐‘˜=๐‘Ÿ(๐›ผ๐‘˜). For this reason, the ๐‘ โ„“ with โ„“mod23โˆˆ๐’ต are called known syndromes. The other ๐‘ โ„“ are called unknown syndromes.

The set of indices ๐‘— for which ๐‘’๐‘—โ‰ 0 is ๐ฟ={๐‘–1,โ€ฆ,๐‘–๐œˆ}. We have 0โ‰ค๐‘–1<๐‘–2<โ‹ฏ<๐‘–๐œˆโ‰ค22. The elements of ๐ฟ are called the error locations, and the ๐‘ง๐‘—=๐›ผ๐‘–๐‘—โˆˆ๐”ฝ311 are the error-location numbers. These are the roots of the error-location polynomial:๐œŽ(๐‘ฅ)=๐‘ฅ๐œˆ+๐œˆโˆ’1๎“๐‘—=0๐œŽ๐œˆโˆ’๐‘—๐‘ฅ๐‘—=๐œˆ๎‘๐‘—=1๎€ท๐‘ฅโˆ’๐‘ง๐‘—๎€ธ,(2)where the ๐œŽ๐‘– are the elementary symmetric functions that in turn are related to the syndromes via Newton's identities [7, pp. 244โ€“245]:๐‘ ๐‘˜+๐œˆ๎“๐‘—=1๐œŽ๐‘—๐‘ ๐‘˜โˆ’๐‘—=0for๐‘˜โˆˆโ„ค.(3)The equations in (3) can be solved efficiently when there are a sufficient number of consecutive known syndromes. However, when decoding QR codes, typically this is not the case. Such difficulty can be overcome by calculating one or more unknown syndromes with the aid of the following result from [2, p. 1182], applied to the nonbinary case [8] (recall that ๐‘ ๐‘˜=โˆ‘๐œˆ๐‘—=1๐‘’๐‘–๐‘—๐‘ง๐‘˜๐‘—, for all ๐‘˜โˆˆโ„ค).

Theorem 1. Let ๐ผ={๐‘–1,๐‘–2,โ€ฆ,๐‘–๐œˆ+1} and ๐ฝ={๐‘—1,๐‘—2,โ€ฆ,๐‘—๐œˆ+1} be two subsets of {0,โ€ฆ,22}. They define two (๐œˆ+1)ร—๐œˆ matrices and one ๐œˆร—๐œˆ diagonal matrix given, respectively, by ๐‘‹๐ผ=โŽกโŽขโŽขโŽขโŽฃ๐‘ง๐‘–11๐‘ง๐‘–12โ‹ฏ๐‘ง๐‘–1๐œˆ๐‘ง๐‘–21๐‘ง๐‘–22โ‹ฏ๐‘ง๐‘–2๐œˆ๐‘งโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘–๐œˆ+11๐‘ง๐‘–๐œˆ+12โ‹ฏ๐‘ง๐‘–๐œˆ+1๐œˆโŽคโŽฅโŽฅโŽฅโŽฆ,๐‘‹๐ฝ=โŽกโŽขโŽขโŽขโŽฃ๐‘ง๐‘—11๐‘ง๐‘—12โ‹ฏ๐‘ง๐‘—1๐œˆ๐‘ง๐‘—21๐‘ง๐‘—22โ‹ฏ๐‘ง๐‘—2๐œˆ๐‘งโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘—๐œˆ+11๐‘ง๐‘—๐œˆ+12โ‹ฏ๐‘ง๐‘—๐œˆ+1๐œˆโŽคโŽฅโŽฅโŽฅโŽฆ,๐‘Œ๐ผ=โŽกโŽขโŽขโŽขโŽฃ๐‘’๐‘–10โ‹ฏ00๐‘’๐‘–2โ‹ฏ0โ‹ฎโ‹ฎโ‹ฑโ‹ฎ00โ‹ฏ๐‘’๐‘–๐œˆโŽคโŽฅโŽฅโŽฅโŽฆ.(4)Then the (๐œˆ+1)ร—(๐œˆ+1) matrix defined by ๐‘†(๐ผ,๐ฝ)=๐‘‹๐ผ๐‘Œ๐ผ๐‘‹๐‘‡๐ฝ is equal to โŽกโŽขโŽขโŽขโŽฃ๐‘ ๐‘†(๐ผ,๐ฝ)=๐‘–1+๐‘—1๐‘ ๐‘–1+๐‘—2โ‹ฏ๐‘ ๐‘–1+๐‘—๐œˆ+1๐‘ ๐‘–2+๐‘—1๐‘ ๐‘–2+๐‘—2โ‹ฏ๐‘ ๐‘–2+๐‘—๐œˆ+1๐‘ โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘–๐œˆ+1+๐‘—1๐‘ ๐‘–๐œˆ+1+๐‘—2โ‹ฏ๐‘ ๐‘–๐œˆ+1+๐‘—๐œˆ+1โŽคโŽฅโŽฅโŽฅโŽฆ.(5)Furthermore, det๐‘†(๐ผ,๐ฝ)=0.

If ๐‘†(๐ผ,๐ฝ) has entries that are unknown syndromes, then Theorem 1 can be used to determine them from the equation det๐‘†(๐ผ,๐ฝ)=0.

3. Calculation of ๐œŽ(๐‘ฅ) for the Ternary (23,11,9) QR Code

In this section the use of Theorem 1 for decoding nonbinary QR codes is illustrated. The focus is on the ternary QR code of length 23 generated by ๐‘”(๐‘ฅ). The final result is an algorithm for finding ๐œŽ(๐‘ฅ), the error-location polynomial, from ๐‘Ÿ(๐‘ฅ). The decoder will determine the coefficients of ๐œŽ(๐‘ฅ), namely, the ๐œŽ๐‘–, from (3). Knowledge of a sequence of consecutive syndromes is required. One choice is ๐‘ 22=๐‘ โˆ’1,๐‘ 0,๐‘ 1,๐‘ 2,๐‘ 3,๐‘ 4,๐‘ 5,๐‘ 6. Observe that any syndrome ๐‘ ๐‘˜ where ๐‘˜โˆˆ๐’ต can be readily computed by the decoder as ๐‘Ÿ(๐›ผ๐‘˜). Since 5โˆ‰๐’ต, ๐‘ 5 is the unknown syndrome to be determined during the decoding procedure described next. Since 5โ‹…9โ‰ก22(mod23), one has ๐‘ โˆ’1=๐‘ 22=๐‘ 95.

Let ๐ผ1={1,2,5,9,21},๐ฝ1={3,4,7,11,22},๐ผ2={0,4,8,19,20}, and ๐ฝ2={4,5,8,12,16}. Form the matrices ๐‘†(๐ผ1,๐ฝ1) and ๐‘†(๐ผ2,๐ฝ2) as in (5),๐‘†๎€ท๐ผ1,๐ฝ1๎€ธ=โŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ 4๐‘ 5๐‘ 8๐‘ 12๐‘ 0๐‘ 5๐‘ 6๐‘ 9๐‘ 13๐‘ 1๐‘ 8๐‘ 9๐‘ 12๐‘ 16๐‘ 4๐‘ 12๐‘ 13๐‘ 16๐‘ 20๐‘ 8๐‘ 1๐‘ 2๐‘ 5๐‘ 9๐‘ 20โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ,๐‘†๎€ท๐ผ2,๐ฝ2๎€ธ=โŽกโŽขโŽขโŽขโŽขโŽฃ๐‘ 4๐‘ 5๐‘ 8๐‘ 12๐‘ 16๐‘ 8๐‘ 9๐‘ 12๐‘ 16๐‘ 20๐‘ 12๐‘ 13๐‘ 16๐‘ 20๐‘ 1๐‘ 0๐‘ 1๐‘ 4๐‘ 8๐‘ 12๐‘ 1๐‘ 2๐‘ 5๐‘ 9๐‘ 13โŽคโŽฅโŽฅโŽฅโŽฅโŽฆ.(6)All the entries in ๐‘†(๐ผ1,๐ฝ1) and ๐‘†(๐ผ2,๐ฝ2) are known except for ๐‘ 5 and ๐‘ 20. However, ๐‘ 20=๐‘ 527. Therefore, ๐‘“1=det๐‘†(๐ผ1,๐ฝ1) and ๐‘“2=det๐‘†(๐ผ1,๐ฝ1) are polynomials in a single variable, namely, ๐‘ 5. The next proposition was verified for each one of the 156906 error patterns of weights 1,2,3, and 4, using Magma [9].Proposition 1. For ๐œˆ=1,2,3,4, gcd(๐‘“1,๐‘“2) is a first-degree polynomial in ๐‘ 5.

The above yields the following procedure for determining ๐œŽ(๐‘ฅ).

Step 1. If ๐‘ 0=๐‘ 1=0, then declare that ๐œˆ=0 and exit. Otherwise, proceed to Step 2.

Step 2. Let ๐‘“=gcd(๐‘“1,๐‘“2). If deg๐‘“=1, solve ๐‘“=0 for ๐‘ 5 and proceed to Step 3. Otherwise, declare that ๐œˆ>4 and exit.

Step 3. Determine the coefficients of the error-location polynomial ๐œŽ(๐‘ฅ) by solving the following linear system for the elementary symmetric functions:๐‘ ๐‘˜=โˆ’๐‘˜โˆ’1๎“๐‘—=๐‘˜โˆ’4๐‘ ๐‘—๐œŽ๐‘˜โˆ’๐‘—for๐‘˜=3,4,5,6.(7)If the linear system is nonsingular and ๐œŽ(๐‘ฅ) has four roots ๐‘ฅ1,โ€ฆ,๐‘ฅ4โˆˆ๐”ฝ311 which satisfy ๐‘ฅ๐‘–23=1 for ๐‘–=1,โ€ฆ,4, then declare ๐œˆ=4 and exit. Otherwise, proceed to Step 4.

Step 4. Solve the following linear system for the elementary symmetric functions:๐‘ ๐‘˜=โˆ’๐‘˜โˆ’1๎“๐‘—=๐‘˜โˆ’3๐‘ ๐‘—๐œŽ๐‘˜โˆ’๐‘—for๐‘˜=4,5,6.(8)If the linear system is nonsingular and ๐œŽ(๐‘ฅ) has three roots ๐‘ฅ1,๐‘ฅ2,๐‘ฅ3โˆˆ๐”ฝ311 which satisfy ๐‘ฅ๐‘–23=1 for ๐‘–=1,2,3, then declare ๐œˆ=3 and exit. Otherwise, proceed to Step 5.

Step 5. Solve the following linear system for the elementary symmetric functions:๐‘ ๐‘˜=โˆ’๐‘˜โˆ’1๎“๐‘—=๐‘˜โˆ’2๐‘ ๐‘—๐œŽ๐‘˜โˆ’๐‘—for๐‘˜=5,6.(9)If the linear system is nonsingular and ๐œŽ(๐‘ฅ) has two roots ๐‘ฅ1,๐‘ฅ2โˆˆ๐”ฝ311 which satisfy ๐‘ฅ๐‘–23=1 for ๐‘–=1,2, then declare ๐œˆ=2 and exit. Otherwise, proceed to Step 6.

Step 6. If we get to this point, then either ๐œˆ=1 or ๐œˆ>4. The coefficient ๐œŽ1 of ๐œŽ(๐‘ฅ) is calculated as ๐œŽ1=โˆ’๐‘ 6/๐‘ 5. If ๐œŽ1โˆˆ๐”ฝ311 is such that ๐œŽ123=1, then ๐œˆ=1. Otherwise, declare that ๐œˆ>4. Exit.


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