Research Letters in Communications

Volume 2009, Article ID 107432, 3 pages

http://dx.doi.org/10.1155/2009/107432

## 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: Guosen Yue

Copyright © 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.

#### Abstract

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 be the set of quadratic residues of and the set of quadratic nonresidues of . The smallest extension of containing , a primitive twenty-third root of unity, is . Denote the set by and define asThe cyclic code generated by 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 be the sent
code polynomial, that is, a multiple of . The received polynomial, denoted by , satisfies where is the error
pattern. Let denote the
Hamming weight of . Observe that can be
correctly determined provided . Only and are known to
the receiver, which seeks to determine the most probable . For any , the syndrome is defined as . It follows that , for all . Observe that for all , whenever . For any , whence . For this reason, the with are called *known* syndromes. The other are called *unknown* syndromes.

The set of indices for which is . We have . The elements of are called the error locations, and the are the error-location numbers. These are the roots of the error-location polynomial:where the are the elementary symmetric functions that in turn are related to the syndromes via Newton's identities [7, pp. 244–245]: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 , for all ).

Theorem 1. *Let and be two subsets
of . They define two matrices and
one diagonal matrix
given, respectively, by **Then the matrix defined
by is equal to **Furthermore, .*

If has entries that are unknown syndromes, then Theorem 1 can be used to determine them from the equation .

#### 3. Calculation of for the Ternary 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 generated by . The final result is an algorithm for finding , the error-location polynomial, from . The decoder will determine the coefficients of , namely, the , from (3). Knowledge of a sequence of consecutive syndromes is required. One choice is . Observe that any syndrome where can be readily computed by the decoder as . Since , is the unknown syndrome to be determined during the decoding procedure described next. Since , one has .

Let , and . Form the matrices and as in
(5),All the entries in and are known
except for and . However, . Therefore, and are polynomials
in a single variable, namely, . The next proposition was verified for each one of
the error patterns
of weights , and , using Magma [9].Proposition 1. *For , is a
first-degree polynomial in .*

The above yields the following procedure for determining .

*Step 1. *If , then declare that and exit.
Otherwise, proceed to Step 2.

*Step 2. *Let . If , solve for and proceed to
Step 3. Otherwise, declare that and exit.

*Step 3. *Determine the coefficients of the
error-location polynomial by solving the
following linear system for the elementary symmetric functions:If the linear system is
nonsingular and has four roots which satisfy for , then declare and exit.
Otherwise, proceed to Step 4.

*Step 4. *Solve the following linear system
for the elementary symmetric functions:If the linear system is
nonsingular and has three roots which satisfy for , then declare and exit.
Otherwise, proceed to Step 5.

*Step 5. *Solve the following linear system
for the elementary symmetric functions:If the linear system is
nonsingular and has two roots which satisfy for , then declare and exit.
Otherwise, proceed to Step 6.

*Step 6. *If we get to this point, then
either or . The coefficient of is calculated
as . If is such that , then . Otherwise, declare that . Exit.

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