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

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 𝒬={1,2,3,4,6,8,9,12,13,16,18} be the set of quadratic residues of 23 and 𝒩 the set of quadratic nonresidues of 23. The smallest extension of 𝔽₃=GF(3) containing 𝛼, a primitive twenty-third root of unity, is 𝔽₃¹¹=GF(3¹¹). Denote the set {0}∪𝒬 by 𝒵 and define 𝑔(𝑥)∈𝔽₃[𝑥] as𝑔(𝑥)==𝑥_{𝑖∈𝒵12}𝑥−𝛼^𝑖+𝑥⁹+𝑥⁷+𝑥⁶+2𝑥⁵+𝑥⁴+2𝑥³+2𝑥+1.(1)The cyclic code generated by 𝑔(𝑥) 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 𝑐(𝑥)=∑^22𝑖=0𝑐_𝑖𝑥^𝑖∈𝔽₃[𝑥] be the sent code polynomial, that is, a multiple of 𝑔(𝑥). The received polynomial, denoted by 𝑟(𝑥)=∑^22𝑖=0𝑟_𝑖𝑥^𝑖, satisfies 𝑟(𝑥)=𝑐(𝑥)+𝑒(𝑥) where 𝑒(𝑥)=∑^22𝑖=0𝑒_𝑖𝑥^𝑖∈𝔽₃[𝑥] is the error pattern. Let 𝜈 denote the Hamming weight of 𝑒(𝑥). Observe that 𝑒(𝑥) can be correctly determined provided 𝜈≤4. Only 𝑔(𝑥) and 𝑟(𝑥) are known to the receiver, which seeks to determine the most probable 𝑒(𝑥). For any 𝑘∈ℤ, the syndrome 𝑠_𝑘 is defined as 𝑠_𝑘=𝑒(𝛼^𝑘). It follows that 𝑠_3𝑘=𝑠^3𝑘, for all 𝑘∈ℤ. Observe that for all 𝑘,ℓ∈ℤ, 𝑠_𝑘=𝑠_ℓ whenever 𝑘≡ℓ(mod23). For any 𝑘∈𝒵,𝑔(𝛼^𝑘)=0, whence 𝑠_𝑘=𝑟(𝛼^𝑘). For this reason, the 𝑠_ℓ with ℓmod23∈𝒵 are called known syndromes. The other 𝑠_ℓ are called unknown syndromes.

The set of indices 𝑗 for which 𝑒_𝑗≠0 is 𝐿={𝑖₁,…,𝑖_𝜈}. We have 0≤𝑖₁<𝑖₂<⋯<𝑖_𝜈≤22. The elements of 𝐿 are called the error locations, and the 𝑧_𝑗=𝛼^𝑖𝑗∈𝔽₃¹¹ are the error-location numbers. These are the roots of the error-location polynomial:𝜎(𝑥)=𝑥^𝜈+^𝜈−1_𝑗=0𝜎_𝜈−𝑗𝑥^𝑗=^𝜈_𝑗=1𝑥−𝑧_𝑗,(2)where the 𝜎_𝑖 are the elementary symmetric functions that in turn are related to the syndromes via Newton's identities [7, pp. 244–245]:𝑠_𝑘+^𝜈_𝑗=1𝜎_𝑗𝑠_𝑘−𝑗=0for𝑘∈ℤ.(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 𝑠_𝑘=∑^𝜈𝑗=1𝑒_𝑖𝑗𝑧^𝑘𝑗, for all 𝑘∈ℤ).

Theorem 1. Let 𝐼={𝑖₁,𝑖₂,…,𝑖_𝜈+1} and 𝐽={𝑗₁,𝑗₂,…,𝑗_𝜈+1} be two subsets of {0,…,22}. They define two (𝜈+1)×𝜈 matrices and one 𝜈×𝜈 diagonal matrix given, respectively, by 𝑋_𝐼=⎡⎢⎢⎢⎣𝑧^𝑖11𝑧^𝑖1₂⋯𝑧^𝑖1_𝜈𝑧^𝑖2₁𝑧^𝑖2₂⋯𝑧^𝑖2_𝜈𝑧⋮⋮⋱⋮_𝑖𝜈+11𝑧^𝑖𝜈+1₂⋯𝑧^𝑖𝜈+1_𝜈𝑋_𝐽⎤⎥⎥⎥⎦,=⎡⎢⎢⎢⎣𝑧^𝑗11𝑧^𝑗1₂⋯𝑧^𝑗1_𝜈𝑧^𝑗2₁𝑧^𝑗2₂⋯𝑧^𝑗2_𝜈𝑧⋮⋮⋱⋮_𝑗𝜈+11𝑧^𝑗𝜈+1₂⋯𝑧^𝑗𝜈+1_𝜈𝑌_𝐼⎤⎥⎥⎥⎦,=⎡⎢⎢⎢⎣𝑒^𝑖10𝑒0_𝑖2⋯0⋮0⋮0⋱⋯⋯𝑒⋮0_𝑖𝜈⎤⎥⎥⎥⎦.(4)Then the (𝜈+1)×(𝜈+1) matrix defined by 𝑆(𝐼,𝐽)=𝑋_𝐼𝑌_𝐼𝑋^𝑇𝐽 is equal to 𝑆(𝐼,𝐽)=⎡⎢⎢⎢⎣𝑠^𝑖1+𝑗1𝑠_𝑖1+𝑗2⋯𝑠_{𝑖1+𝑗𝜈+1}𝑠_𝑖2+𝑗1𝑠_𝑖2+𝑗2⋯𝑠_{𝑖2+𝑗𝜈+1}𝑠⋮⋮⋱⋮_{𝑖𝜈+1+𝑗1}𝑠_{𝑖𝜈+1+𝑗2}⋯𝑠_{𝑖𝜈+1+𝑗𝜈+1}⎤⎥⎥⎥⎦.(5)Furthermore, det𝑆(𝐼,𝐽)=0.

If 𝑆(𝐼,𝐽) has entries that are unknown syndromes, then Theorem 1 can be used to determine them from the equation det𝑆(𝐼,𝐽)=0.

3. Calculation of 𝜎(𝑥) 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 𝑔(𝑥). The final result is an algorithm for finding 𝜎(𝑥), the error-location polynomial, from 𝑟(𝑥). The decoder will determine the coefficients of 𝜎(𝑥), namely, the 𝜎_𝑖, from (3). Knowledge of a sequence of consecutive syndromes is required. One choice is 𝑠₂₂=𝑠₋₁,𝑠₀,𝑠₁,𝑠₂,𝑠₃,𝑠₄,𝑠₅,𝑠₆. Observe that any syndrome 𝑠_𝑘 where 𝑘∈𝒵 can be readily computed by the decoder as 𝑟(𝛼^𝑘). Since 5∉𝒵, 𝑠₅ is the unknown syndrome to be determined during the decoding procedure described next. Since 5⋅9≡22(mod23), one has 𝑠₋₁=𝑠₂₂=𝑠⁹⁵.

Let 𝐼₁={1,2,5,9,21},𝐽₁={3,4,7,11,22},𝐼₂={0,4,8,19,20}, and 𝐽₂={4,5,8,12,16}. Form the matrices 𝑆(𝐼₁,𝐽₁) and 𝑆(𝐼₂,𝐽₂) as in (5),𝑆𝐼₁,𝐽₁=⎡⎢⎢⎢⎢⎣𝑠⁴𝑠₅𝑠₈𝑠₁₂𝑠₀𝑠₅𝑠₆𝑠₉𝑠₁₃𝑠₁𝑠₈𝑠₉𝑠₁₂𝑠₁₆𝑠₄𝑠₁₂𝑠₁₃𝑠₁₆𝑠₂₀𝑠₈𝑠₁𝑠₂𝑠₅𝑠₉𝑠₂₀𝑆𝐼₂⎤⎥⎥⎥⎥⎦,,𝐽₂=⎡⎢⎢⎢⎢⎣𝑠⁴𝑠₅𝑠₈𝑠₁₂𝑠₁₆𝑠₈𝑠₉𝑠₁₂𝑠₁₆𝑠₂₀𝑠₁₂𝑠₁₃𝑠₁₆𝑠₂₀𝑠₁𝑠₀𝑠₁𝑠₄𝑠₈𝑠₁₂𝑠₁𝑠₂𝑠₅𝑠₉𝑠₁₃⎤⎥⎥⎥⎥⎦.(6)All the entries in 𝑆(𝐼₁,𝐽₁) and 𝑆(𝐼₂,𝐽₂) are known except for 𝑠₅ and 𝑠₂₀. However, 𝑠₂₀=𝑠⁵²⁷. Therefore, 𝑓₁=det𝑆(𝐼₁,𝐽₁) and 𝑓₂=det𝑆(𝐼₁,𝐽₁) are polynomials in a single variable, namely, 𝑠₅. 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 𝜈=1,2,3,4, gcd(𝑓₁,𝑓₂) is a first-degree polynomial in 𝑠₅.

The above yields the following procedure for determining 𝜎(𝑥).

Step 1. If 𝑠₀=𝑠₁=0, then declare that 𝜈=0 and exit. Otherwise, proceed to Step 2.

Step 2. Let 𝑓=gcd(𝑓₁,𝑓₂). If deg𝑓=1, solve 𝑓=0 for 𝑠₅ and proceed to Step 3. Otherwise, declare that 𝜈>4 and exit.

Step 3. Determine the coefficients of the error-location polynomial 𝜎(𝑥) by solving the following linear system for the elementary symmetric functions:𝑠_𝑘=−^𝑘−1_𝑗=_𝑘−4𝑠_𝑗𝜎_𝑘−𝑗for𝑘=3,4,5,6.(7)If the linear system is nonsingular and 𝜎(𝑥) has four roots 𝑥₁,…,𝑥₄∈𝔽₃¹¹ which satisfy 𝑥^𝑖23=1 for 𝑖=1,…,4, then declare 𝜈=4 and exit. Otherwise, proceed to Step 4.

Step 4. Solve the following linear system for the elementary symmetric functions:𝑠_𝑘=−^𝑘−1_𝑗=_𝑘−3𝑠_𝑗𝜎_𝑘−𝑗for𝑘=4,5,6.(8)If the linear system is nonsingular and 𝜎(𝑥) has three roots 𝑥₁,𝑥₂,𝑥₃∈𝔽₃¹¹ which satisfy 𝑥^𝑖23=1 for 𝑖=1,2,3, then declare 𝜈=3 and exit. Otherwise, proceed to Step 5.

Step 5. Solve the following linear system for the elementary symmetric functions:𝑠_𝑘=−^𝑘−1_𝑗=_𝑘−2𝑠_𝑗𝜎_𝑘−𝑗for𝑘=5,6.(9)If the linear system is nonsingular and 𝜎(𝑥) has two roots 𝑥₁,𝑥₂∈𝔽₃¹¹ which satisfy 𝑥^𝑖23=1 for 𝑖=1,2, then declare 𝜈=2 and exit. Otherwise, proceed to Step 6.

Step 6. If we get to this point, then either 𝜈=1 or 𝜈>4. The coefficient 𝜎₁ of 𝜎(𝑥) is calculated as 𝜎₁=−𝑠₆/𝑠₅. If 𝜎₁∈𝔽₃¹¹ is such that 𝜎¹²³=1, then 𝜈=1. Otherwise, declare that 𝜈>4. Exit.