(Open Access) An Interpolation Procedure for List Decoding Reed–Solomon Codes Based on Generalized Key Equations (2011) | Alexander Zeh

Q: What have the authors stated for future works in "An interpolation procedure for list decoding reed–solomon codes based on generalized key equations" ?

The authors reformulated the Guruswami–Sudan interpolation conditions ( for a multiplicity higher than one ) for Generalized Reed–Solomon codes into a set of univariate polynomial equations, which can partially be seen as Extended Key Equations. The authors adapted the Fundamental Iterative Algorithm of Feng and Tzeng to this special structure and achieved a significant reduction of the time complexity. As mentioned in Note 2, the set of equations can be further reduced, under the observation that the diagonal terms are constant, i. e., they do not depend on the received word.

5946 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011

An Interpolation Procedure for List Decoding

Reed–Solomon Codes Based on

Generalized Key Equations

Alexander Zeh, Student Member, IEEE, Christian Gentner, Member, IEEE, and Daniel Augot

Abstract—The key step of syndrome-based decoding of

Reed–Solomon codes up to half the minimum distance is to

solve the so-called Key Equation. List decoding algorithms, ca-

pable of decoding beyond half the minimum distance, are based

on interpolation and factorization of multivariate polynomials.

This article provides a link between syndrome-based decoding ap-

proaches based on Key Equations and the interpolation-based list

decoding algorithms of Guruswami and Sudan for Reed–Solomon

codes. The original interpolation conditions of Guruswami and

Sudan for Reed–Solomon codes are reformulated in terms of

a set of Key Equations. These equations provide a structured

homogeneous linear system of equations of Block-Hankel form,

that can be solved by an adaption of the Fundamental Iterative

Algorithm. For an

(

n; k

)

Reed–Solomon code, a multiplicity

and

a list size

, our algorithm has time complexity

(

)

Index Terms—Block-Hankel matrix, fundamental iterative algo-

rithm (FIA), Guruswami–Sudan interpolation, key equation, list

decoding, Reed–Solomon codes.

I. INTRODUCTION

N 1999, Guruswami and Sudan [3]–[5] extended Sudan’s

original approach [6] by introducing multiplicities in the

interpolation step of their polynomial-time list decoding proce-

dure for Reed–Solomon and Algebraic Geometric codes. This

modiﬁcation permits decoding of

Reed–Solomon codes

[7] (and Algebraic Geometric codes) of arbitrary code-rate

with increased decoding radius. Guruswami and

Sudan were focused on the existence of a polynomial-time

algorithm. Kötter [8] and Roth-Ruckenstein [9], [10] pro-

posed quadratic time algorithms for the key steps of the

Guruswami–Sudan principle for Reed–Solomon codes, i.e.,

Manuscript received September 03, 2009; revised September 13, 2010; ac-

cepted April 10, 2011. Date of current version August 31, 2011. This work

was supported by the German Research Council “Deutsche Forschungsgemein-

schaft” (DFG) by Grant No. Bo867/22-1. The material in this paper was pre-

sented at the IEEE International Symposium on Information Theory, Toronto,

ON, Canada, 2008, and at the IEEE Information Theory Workshop, Taormina,

Sicily, Italy, 2009.

A. Zeh is with the Institute of Telecommunications and Applied In-

formation Theory, University of Ulm, Germany, and also with the

INRIA—Saclay-Île-de-France and École Polytechnique, Paris, France

(e-mail: alexander.zeh@uni-ulm.de).

C. Gentner is with the Institute of Communications and Navigation of the

German Aerospace Center (DLR), Germany (e-mail: christian.gentner@dlr.de).

D. Augot is with INRIA—Saclay-Île-de-France and École Polytechnique,

Paris, France (e-mail: daniel.augot@inria.fr).

Communicated by M. Blaum, Associate Editor for Coding Theory.

Digital Object Identiﬁer 10.1109/TIT.2011.2162160

interpolation and factorization of bivariate polynomials. Var-

ious other approaches for a low-complexity realization of

Guruswami–Sudan exist, e.g., the work of Alekhnovich [11],

where fast computer algebra techniques are used. Trifonov’s

[12] contributions rely on ideal theory and divide and conquer

methods. Sakata uses Gröbner-bases techniques [13], [14].

In this paper, we reformulate the

bivariate interpolation step

of Guruswami–Sudan for Reed–Solomon codes in a set of uni-

variate Key Equations [1]. This extends the previous work of

Roth and Ruckenstein [9], [10], where the reformulation was

done for the special case of Sudan. Furthermore, we present a

modiﬁcation of the so-called Fundamental Iterative Algorithm

(FIA), proposed by Feng and Tzeng in 1991 [15]. Adjusted to

the special case of one Hankel matrix the FIA resembles the ap-

proach of Berlekamp and Massey [16], [17].

Independently of our contribution, Beelen and Høholdt refor-

mulated the Guruswami–Sudan constraints for Algebraic Geo-

metric codes [18], [19]. It is not clear, if the system they obtain

is highly structured.

This contribution is organized as follows. The next section

contains basic deﬁnitions for Reed–Solomon codes and bi-

variate polynomials. In Section III, we derive the Key Equation

for conventional decoding of Reed–Solomon codes from the

Welch-Berlekamp approach [20] and we present the adjust-

ment of the FIA for one Hankel matrix. A modiﬁed version

of Sudan’s reformulated interpolation problem based on the

work of Roth-Ruckenstein [9] is derived and the adjustment of

the FIA for this case is illustrated in Section IV. In Section V,

the interpolation step of the Guruswami–Sudan principle is

reformulated. The obtained homogeneous set of linear equa-

tions has Block-Hankel structure. We adjust the FIA for this

Block-Hankel structure, prove the correctness of the proposed

algorithm and analyze its complexity. We conclude this contri-

bution in Section VI.

II. D

EFINITIONS AND

PRELIMINARIES

Throughout this paper,

denotes the set of inte-

gers

and denotes the set of integers

.An matrix consists

of the entries

, where and . A uni-

variate polynomial

of degree less than is denoted by

. A vector of length is represented by

Let

be a power of a prime and let denote the

ﬁnite ﬁeld of order

. Let denote nonzero dis-

tinct elements (code-locators) of

and let denote

ZEH et al.: INTERPOLATION PROCEDURE FOR LIST DECODING REED–SOLOMON CODES 5947

nonzero elements (column-multipliers), the associated evalua-

tion map ev is

(1)

The associated Generalized Reed–Solomon code

length

and dimension is [21]

(2)

where

denotes the set of all univariate polynomials with

degree less than

. Generalized Reed–Solomon codes are MDS

codes with minimum distance

. The dual of a Gen-

eralized Reed–Solomon is also a Generalized Reed–Solomon

code with the same code locators and column multipliers

, where . The

explicit form of the column multipliers is [22]

(3)

We will take advantage of structured matrices and therefore

we recall the deﬁnition of a Hankel matrix in the following.

Deﬁnition 1 (Hankel Matrix): An

Hankel matrix

is a matrix, where for all

and holds.

Let us recall some properties of bivariate polynomials in

Deﬁnition 2 (Weighted Degree): Let the polynomial

be in . Then, the

-weighted degree of , denoted by ,

is the maximum over all

such that .

Deﬁnition 3 (Multiplicity and Hasse Derivative [23]): Let

be a polynomial in . Let

. A bivariate

polynomial

has at least multiplicity in the point

, denoted by

(4)

if the coefﬁcients

are zero for all . Furthermore,

the

th Hasse derivative of the polynomial in the

point

(5)

Let denote the th Hasse derivative of

with respect to the variable .

We will use the inner product for bivariate polynomials to

describe our algorithms.

Deﬁnition 4 (Inner Product): Let two polynomials

and

in be given. The inner product of

and is deﬁned by .

III. W

ELCH-BERLEKAMP AS

LIST-ONE DECODER AND THE

FUNDAMENTAL ITERATIVE

ALGORITHM

A. Syndrome-Based Decoding of Reed–Solomon Codes

Let

denote the error word and let

be the set of error locations (that is ). Let

. It is well known that a code can

recover uniquely any error pattern if and only if

. The

syndrome coefﬁcients depend only

on the error word

and the associated syndrome polynomial

is deﬁned by [22]

The error-locator polynomial is and the

error-evaluator polynomial

. They are related by the Key Equation:

(6)

The main steps for conventional decoding up to half the min-

imum distance are:

1) Calculate the syndrome polynomial

from the re-

ceived word

2) Solve (6) for the error-locator polynomial

and deter-

mine its roots.

3) Compute

and then determine the error values.

B. Derivation of the Key Equation From Welch-Berlekamp

We derive the classical Key Equation (6) from the simplest in-

terpolation based decoding algorithm, reported as the “Welch-

Berlekamp” decoding algorithm in [24]–[26]. We provide a sim-

pler representation than in [20] and give a polynomial derivation

of the Key Equation.

Consider a

code with support set ,

multipliers

and dimension . The Welch-

Berlekamp approach is based on the following lemma [27, Ch.

5.2].

Lemma 1 (List-One Decoder): Let

be a code-

word of a

code and let

be the received word. We search for a polynomial

in such that:

has distance less than or equal to from the

received word

, then .

Let us connect Lemma 1 to (6).

Proposition 1 (Univariate Reformulation): Let

be the Lagrange interpolation polynomial, such that

holds. Let .

Then

satisﬁes Conditions 2)

5948 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011

and 3) of Lemma 1 if and only if there exists a polynomial

such that

(7)

and

Let

. Deﬁne the following

reciprocal polynomials:

(8)

Inverting the order of the coefﬁcients of (7) leads to:

With (8), we obtain:

which we can consider modulo . We obtain

(9)

Since

, we can deﬁne the formal power series

(10)

Using the column multipliers (3) for the dual code, it can be

veriﬁed that

is the series of syndromes with

(11)

Thus, dividing (9) by

, we obtain

(12)

which corresponds to the classical Key Equation (6). The syn-

drome polynomial is

, and is the error-

locator polynomial

In the case of

errors, we consider only the terms of the

Key Equation of degree greater than

and we get the

following homogeneous linear system of equations:

(13)

The above syndrome matrix

for all and

has Hankel form (see Deﬁnition 1). Equation (12)

can be solved by the well-known Berlekamp-Massey algorithm

[16], [17] or with a modiﬁcation of the Extended Euclidean al-

gorithm [28]. The parallels of the Berlekamp-Massey algorithm

and the Extended Euclidean algorithm have been considered in

[29]–[31].

We consider in the following the FIA [15], that can be used

to ﬁnd the ﬁrst

linearly dependent columns and connec-

tion coefﬁcients

for an arbitrary matrix. The FIA

allows a signiﬁcant reduction of complexity when adjusted to a

Hankel matrix as in (13).

C. The FIA for One Hankel Matrix

Given an arbitrary

matrix , the FIA

outputs the minimal number of

linearly dependent

columns together with the polynomial

with

, such that holds.

The FIA scans the

th column of the matrix row-wise in the

order

and uses previously stored polynomials to

update the current polynomial

. Let be the index of the

current column under inspection, and let

be the current candidate polynomial that satisﬁes

for some value of the row index . In other words, the coefﬁ-

cients of the polynomial

give us the vanishing linear com-

bination of the matrix consisting of the ﬁrst

rows and the ﬁrst

columns of the matrix . Suppose that the discrepancy

(14)

for next row

is nonzero. If there exists a previously stored

polynomial

and a nonzero discrepancy , corre-

sponding to row

, then the current polynomial is updated

in the following way:

(15)

The proof of the above update rule is straightforward [15].

In the case

and there is no discrepancy stored,

the actual discrepancy

is stored as . The corresponding

auxiliary polynomial is stored as

. Then, the FIA exam-

ines a new column

Deﬁnition 5 (True Discrepancy): Let the FIA examine the

th row of the th column of matrix . Furthermore, let the

calculated discrepancy (14) be nonzero and no other nonzero

discrepancy be stored for row

. Then, the FIA examines a new

column

. We call this case a true discrepancy.

Theorem 1 (Correctness and Complexity of the FIA [15]):

For an

matrix with , the Fundamental Iterative

Algorithm stops, when the row pointer has reached the last row

ZEH et al.: INTERPOLATION PROCEDURE FOR LIST DECODING REED–SOLOMON CODES 5949

Fig. 1. Illustration of the row pointer



of the classic FIA [(a)] and of the adjusted FIA [b)] when both algorithms are applied to the same 6

7 Hankel syndrome

matrix of a

GRS

(16

;

code. The dots indicate a true discrepancy. In this case, both algorithms enter a new column, but with different initial values of their row

pointers.

of column . Then, the last polynomial corresponds to

a valid combination of the ﬁrst

columns. The complexity

of the algorithm is

For a Hankel matrix

(as in Deﬁnition 1), the FIA can be

adjusted. Assume the case of a true discrepancy, when the FIA

examines the

th row of the th column of the structured matrix

. The current polynomial is . Then, the FIA starts exam-

ining the

th column at row with

and not at row zero. This reduces the cubic time complexity into

a quadratic time complexity [15].

To illustrate the complexity reduction of the FIA when ad-

justed to a Hankel matrix (compared to the original, unadjusted

FIA), we traced the examined rows for each column in Fig. 1.

Fig. 1(a) shows the values of

of the FIA without any adaption.

The row pointer

of the adapted FIA is traced in Fig. 1(b).

The points on the lines in both ﬁgures indicate the case, where

a true discrepancy has been encountered.

IV. S

UDAN INTERPOLATION STEP WITH A HORIZONTAL BAND

OF HANKEL MATRICES

A. Univariate Reformulation of the Sudan Interpolation Step

In this section, we recall parts of the work of Roth and Ruck-

enstein [9], [10] for the interpolation step of the Sudan [6] prin-

ciple. The aimed decoding radius is denoted by

, the corre-

sponding list size is

Problem 1 (Sudan Interpolation Step [6]): Let the aimed

decoding radius

and the received word be

given. The Sudan interpolation step determines a polynomial

, such that

;

We present here a slightly modiﬁed version of [9], to get an

appropriate basis for the extension to the interpolation step in

the Guruswami–Sudan case.

We have

. Let be the Lagrange interpolation polynomial,

s.t.

and .

The reciprocal polynomial of

is denoted by

Similar to Proposition 1, Roth-Ruckenstein [9] proved the

following. There is an interpolation polynomial

satis-

fying Conditions (2) and (3) if and only if there exists a uni-

variate polynomial

with degree smaller than ,

s.t.

Let the reciprocal polynomials be deﬁned as in (8). From [9,

(19)] we have

(16)

where

. We introduce the power series

(17)

Inserting (17) into (16) leads to

(18)

Based on (18) we can now deﬁne syndromes for Problem 1.

Deﬁnition 6 (Syndromes for Sudan): The

generalized

syndrome polynomials

are given

(19)

The ﬁrst-order Extended Key Equation is

(20)

5950 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 9, SEPTEMBER 2011

with .

An explicit form of

(21)

Note 1: In [9], a further degree reduction is proposed. Then

(18), is modulo

and the polynomial disappears.

We do not present this improvement here, because we cannot

properly reproduce this behavior in the Guruswami–Sudan case

(see Note 2).

The degree of the LHS of (16) is smaller than

If we consider the terms of degree higher than

,we

obtain

homogeneous linear equations. Reverting back to the

originals univariate polynomials

, we get the following

system:

(22)

With

, we obtain the fol-

lowing matrix form:

(23)

where each submatrix

is a Hankel matrix. The syndrome polynomials

of Deﬁnition 6 are associated with

this horizontal band of

Hankel matrices by .

In the following, we describe how the FIA can be adapted to

solve the homogeneous system of (23).

B. Adjustment of the FIA for the Reformulated Sudan

Interpolation Problem

The FIA can directly be applied to the matrix

of (23), but if we want to take ad-

vantage of the Hankel structure we have to scan the columns of

in a manner given by the weighted degree

requirement of the interpolation problem.

Let

denote the ordering for the pairs

, where is given by

(24)

The pair that immediately follows

with respect to

the order deﬁned by

is denoted by . The

columns of the matrix

are reordered

according to

. The pair indexes the th column of

th submatrix . More explicitly, we obtain the following

matrix

, where the columns of are reordered [see (25) at

the bottom of the page].

The corresponding homogeneous system of equations can

now be written in terms of the inner product for bivariate poly-

nomials (see Deﬁnition 4).

Problem 2 (Reformulated Sudan Interpolation Problem): Let

the

syndrome polynomials

be given by Deﬁnition 6 and let be

the corresponding bivariate syndrome polynomial. We search a

nonzero bivariate polynomial

such that

(26)

Hence, the bivariate polynomial

is a valid interpo-

lation polynomial for Problem 1. Note that each polynomial

, as deﬁned in (16), has degree smaller than .

To index the columns of the rearranged matrix

, let

(27)

Algorithm 1 is the modiﬁed FIA for solving Problem 2. In con-

trast to the original Roth-Ruckenstein adaption we consider all

homogeneous linear equations (instead of ), according to Note

1. The column pointer is given by

, for indexing the th

column of the

th submatrix . Algorithm 1 virtually scans

the rearranged matrix

column after column (see Line 23 of

Algorithm 1). The true discrepancy value for row

is stored in

array

as , and the corresponding intermediate bivariate

polynomial is stored in array

as . The discrepancy calcu-

lation and the update rule [see (14) and (15) for the basic FIA] is

adapted to the bivariate case (see Line 16 of Algorithm 1). For

each submatrix

, the previous value of the row pointer is

(25)

An Interpolation Procedure for List Decoding Reed–Solomon Codes Based on Generalized Key Equations

Figures

Citations

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Computing minimal interpolation bases

On Rational Interpolation-Based List-Decoding and List-Decoding Binary Goppa Codes

List Decoding of Algebraic Codes

Bases of relations in one or several variables : fast algorithms and applications

References

Polynomial Codes Over Certain Finite Fields

Algebraic Coding Theory

Shift-register synthesis and BCH decoding

Improved decoding of Reed-Solomon and algebraic-geometry codes

Decoding of Reed Solomon Codes beyond the Error-Correction Bound

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