Goppa and Srivastava codes over finite rings

Goppa and Srivastava codes over finite rings

Antonio Aparecido de Andrade^I; Reginaldo Palazzo Jr.^II,*

^IDepartment of Mathematics, Ibilce, Unesp, 15054-000 São José do Rio Preto, SP, Brazil, E-mail: [e-mail]

^IIDepartment of Telematics, Feec, Unicamp, 13083-852 Campinas, SP, Brazil, E-mail: [e-mail]

ABSTRACT

Goppa and Srivastava codes over arbitrary local finite commutative rings with identity are constructed in terms of parity-cleck matrices. An efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, is proposed for Goppa codes.

Mathematical subject classification: 11T71, 94B05, 94B40.

Key words: Goppa code, Srivastava code.

1 Introduction

Goppa codes form a subclass of alternant codes and they are described in terms of a polynomial called Goppa polynomial. The most famous subclasses of alternant codes are BCH codes and Goppa codes, the former for their simple and easily instrumented decoding algorithm, and the latter for meeting the Gilbert-Varshamov bound. However, most of the work regarding construction and decoding of Goppa codes has been done considering codes over finite fields. On the other hand, linear codes over rings have recently generated a great deal of interest.

Linear codes over local finite commutative rings with identity have been discussed in papers by Andrade [1], [2], [3] where it was extended the notion of Hamming, Reed-Solomon, BCH and alternant codes over these rings.

In this paper we describe a construction technique of Goppa and Srivastava codes over local finite commutative rings. The core of the construction technique mimics that of Goppa codes over a finite field, and is addressed, in this paper, from the point of view of specifying a cyclic subgroup of the group of units of an extension ring of finite rings. The decoding algorithm for Goppa codes consists of four major steps: (1) calculation of the syndromes, (2) calculation of the elementary symmetric functions by modified Berlekamp-Massey algorithm, (3) calculation of the error-location numbers, and (4) calculation of the error magnitudes.

This paper is organized as follows. In Section 2, we describe a construction of Goppa codes over local finite commutative rings and an efficient decoding procedure. In Section 3, we describe a construction of Srivastava codes over local finite commutative rings. Finally, in Section 5, the concluding remarks are drawn.

2 Goppa Codes

In this section we describe a construction technique of Goppa codes over arbitrary local finite commutative rings in terms of parity-check matrices, which is very similar to the one proposed by Goppa [4] over finite fields. First, we review basic facts from the Galois theory of local finite commutative rings.

Throughout this paper denotes a local finite commutative ring with identity, maximal ideal and residue field = º GF(p^m), for some prime p, m will be a positive integer, and [x] denotes the ring of polynomials in the variable x over . The natural projection [x] ® [x] is denoted by µ, where µ(a(x)) = (x).

Let f(x) be a monic polynomial of degree h in [x] such that µ(f(x)) is irreducible in [x]. Then f(x) is also irreducible in [x] [5, Theorem XIII.7]. Let be the ring [x]/áf(x)ñ. Then is a local finite commutative ring with identity and it is called a Galois extension of of degree h. Its residue field is ₁ = / º GF(p^mh), where is the unique maximal ideal of , and is the multiplicative group of ₁, whose order is p^mh-1.

Let * denote the multiplicative group of units of . It follows that * is an abelian group, and therefore it can be expressed as a direct product of cyclic groups. We are interested in the maximal cyclic subgroup of *, hereafter denoted by _s, whose elements are the roots of x^s - 1 for some positive integer s such that gcd(s, p) = 1. There is only one maximal cyclic subgroup of * having order relatively prime to p [5, Theorem XVIII.2]. This cyclic group has order s = p^mh-1.

The Goppa codes are specified in terms of a polynomial g(z) called Goppa polynomial. In contrast to cyclic codes, where it is difficult to estimate the minimum Hamming distance d from the generator polynomial, Goppa codes have the property that d > deg(g(z)) + 1.

Let g(z) = g₀ + g₁z + ¼ + g_rz^r be a polynomial with coefficients in and g_r ¹ 0. Let h = (a₁, a₂, ¼ , a_n) be a vector consisting of distinct elements of _s such that g(a_i) are units from for i = 1, 2, ¼ , n.

Definition 2.1. A shortened Goppa code (h, w, g) of length n < s over

has parity-check matrix

where w = (g(), g(), ¼, g()). The polynomial g(z) is called Goppa polynomial.

Definition 2.2. Let (h, w, g) be a Goppa code.

• If g(z) is irreducible then (h, w, g) is called an irreducible Goppa code.

• If, for all c = (c₁, c₂, ¼, c_n) Î (h, w, g), it is true that c¢ = (c_n, c_n-1, ¼, c₁) Î (h, w, g), then (h, w, g) is called a reversible Goppa code.

• If g(z) = (z - a)^r then (h, w, g) is called a commutative Goppa code.

• If g(z) has no multiple zeros then (h, w, g) is called a separable Goppa code.

Remark 2.1. Let (h, w, g) be a Goppa code.

1. We have (h, w, g) is a linear code.

2. A parity-check matrix with elements from is then obtained by replacing each entry of H by the corresponding column vector of length h from .

3. For a Goppa code with polynomial g_l(z) = (z - b_l)^rl, where b_l Î _s, we have

which is row-equivalent to

Consequently, if g(z) = (z - b_l)^rl = g_l(z), then the Goppa code is the intersection of Goppa codes with Goppa polynomial g_l(z) = (z - b_l)^rl, for l = 1, 2, ¼, k, and its parity-check matrix is given by

4. Alternant codes are a special case of Goppa codes [3, Definition 2.1].

It is possible to obtain an estimate of the minimum Hamming distance d of (h, w, g) directly from the Goppa polynomial g(z). The next theorem provides such an estimate.

Theorem 2.1. The code (h, w, g) has minimum Hamming distance d > r + 1.

Proof. We have (h, w, g) is an alternant code (h, w, g) with h = (a₁, a₂, ¼, a_n) and w = (g(a₁)^-1, g(a₂)^-1, ¼, g(a_n)^-1) [3, Definition 2.1]. By [3, Theorem 2.1] it follows (h, w, g) has minimum distance d > r + 1.

Example 2.1. Let = ₂[i] and = , where f(x) = x³ + x + 1 is irreducible over and i² = -1. If a is a root of f(x), then a generates a cyclic group _s of order s = 2³ - 1 = 7. Let h = (a, a⁴, 1, a²), g(z) = z³ + z² + 1 and w = (g(a)^-1, g(a⁴)^-1, g(1)^-1, g(a²)^-1) = (a³, a⁵, 1, a⁶). Since deg(g(z)) = 3, it follows that

is the parity-check matrix of a Goppa code (h, w, g) over with length 4 and minimum Hamming distance at least 4.

Example 2.2. Let = ₂[i] and = , where f(x) = x⁴ + x + 1 is irreducible over . Thus s = 15 and ₁₅ is generated by a, where a⁴ = a + 1. Let g(z) = z⁴ + z³ + 1, h = (1, a, a², a³, a⁴, a⁵, a⁶, a⁸, a⁹, a¹⁰, a¹²) and w = (1, a⁶, a¹², a¹³, a⁹, a¹⁰, a¹¹, a³, a¹⁴, a⁵, a⁷). Since deg(g(z)) = 4, it follows that

is the parity-check matrix of a Goppa code (h, w, g) over ₂[i] with length 11 and minimum Hamming distance at least 5.

2.1 Decoding procedure

In this subsection we present a decoding algorithm for Goppa codes (h, w, g). This algorithm is based on the modified Berlekamp-Massey algorithm [6] which corrects all errors up to the Hamming weight t < r/2, i.e., whose minimum Hamming distance is r + 1.

We first establish some notation. Let be a local finite commutative ring with identity as defined in Section 2 and a be a primitive element of the cyclic group _s, where s = p^mh - 1. Let c = (c₁, c₂, ¼, c_n) be a transmitted codeword and b = (b₁, b₂, ¼, b_n) be the received vector. Thus the error vector is given by e = (e₁, e₂, ¼, e_n) = b - c.

Given a vector h = (a₁, a₂, ¼, a_n) = () in , we define the syndrome values s_l of an error vector e = (e₁, e₂, ¼, e_n) as

Suppose that n < t is the number of errors which occurred at locations x₁ = with values .

Since s = (s₀, s₁, ¼, s_r-1) = bH^t = eH^t, then the first r syndrome values s_l can be calculated from the received vector b as

The elementary symmetric functions s₁, s₂, ¼, s_n of the error-location numbers x₁, x₂, ¼, x_n are defined as the coefficients of the polynomial

where s₀ = 1. Thus, the decoding algorithm being proposed consists of four major steps:

Step 1 - Calculation of the syndrome vector s from the received vector.

Step 2 - Calculation of the elementary symmetric functions s₁, s₂, ¼, s_n from s, using the modified Berlekamp-Massey algorithm [6].

Step 3 - Calculation of the error-location numbers x₁, x₂, ¼, x_n from s₁, s₂, ¼, s_n, that are roots of s(x).

Step 4 - Calculation of the error magnitudes y₁, y₂, ¼, y_n from x_i and s, using Forney's procedure [7].

Now, each step of the decoding algorithm is analyzed. There is no need to comment on Step 1 since the calculation of the vector syndrome is straightforward. The set of possible error-location numbers is a subset of {a⁰, a¹, ¼, a^s-1}. In Step 2, the calculation of the elementary symmetric functions is equivalent to finding a solution s₁, s₂, ¼, s_n, with minimum possible n, to the following set of linear recurrent equations over

where s₀, s₁, ¼, s_r-1 are the components of the syndrome vector. We make use of the modified Berlekamp-Massey algorithm to find the solutions of Equation (2). The algorithm is iterative, in the sense that the following n - l_n equations (called power sums)

are satisfied with l_n as small as possible and = 1. The polynomial s⁽ⁿ⁾(x) = represents the solution at the n-th stage. The n-th discrepancy is denoted by d_n and defined by d_n = . The modified Berlekamp-Massey algorithm for commutative rings with identity is formulated as follows. The inputs to the algorithm are the syndromes s₀, s₁, ¼, s_r-1 which belong to . The output of the algorithm is a set of values s_i, i = 1, 2, ¼, n, such that Equation (2) holds with minimum n. Let s^(-1)(x) = 1, l_-1 = 0, d_-1 = 1, s⁽⁰⁾(x) = 1, l₀ = 0 and d₀ = s₀ be the a set of initial conditions to start the algorithm as in Peterson [8]. The steps of the algorithm are:

1. n ¬ 0.

2. If d_n = 0, then s⁽ⁿ⁺¹⁾(x) ¬ s⁽ⁿ⁾(x) and l_n+1¬ l_n and to go 5).

3. If d_n ¹ 0, then find m < n - 1 such that d_n - yd_m = 0 has a solution y and m - l_m has the largest value. Then, s⁽ⁿ⁺¹⁾(x) ¬ s⁽ⁿ⁾(x) - yx^n-ms^(m)(x) and l_n+1 ¬ max{l_n, l_m + n - m}.

4. If l_n+1 = max{l_n, n + 1 - l_n} then go to step 5, else search for a solution D⁽ⁿ⁺¹⁾(x) with minimum degree l in the range max{l_n, n + 1 - l_n} < l < l_n+1 such that s^(m)(x) defined by D⁽ⁿ⁺¹⁾(x) - s⁽ⁿ⁾(x) = x^n-ms^(m)(x) is a solution for the first m power sums, d_m = -d_n, with a zero divisor in . If such a solution is found, s⁽ⁿ⁺¹⁾(x) ¬ D⁽ⁿ⁺¹⁾(x) and l_n+1¬ l.

5. If n < r - 1, then d_n = s_n + .

6. n ¬ n + 1; if n < r - 1 go to 2); else stop.

The coefficients satisfy Equation (2). At Step 3, the solution to Equation (2) is generally not unique and the reciprocal polynomial r(z) of the polynomial s^(r)(z) (output by the modified Berlekamp-Massey algorithm), may not be the correct error-locator polynomial

(z - x₁)(z - x₂)¼(z - x_n),

where x_j = , for j = 1, 2, ¼, n and i = 1, 2, ¼, n, are the correct error-location numbers. Thus, the procedure for the calculation of the correct error-location numbers is the following:

• compute the roots

  z

  ₁,

  z

  ₂, ¼,

  z

  _n of r(

  z);

among the x_i = , j = 1, 2 ¼, n, select those x_i's such that x_i - z_i are zero divisors in . The selected x_i's will be the correct error-location numbers and each k_j, for j = 1, 2, ¼, n, indicates the position j of the error in the codeword.

At Step 4, the calculation of the error magnitude is based on Forney's procedure [7]. The error magnitude is given by

for j = 1, 2, ¼, n, where the coefficients s_jl are recursively defined by

s_j,i = s_i + x_js_j,i-1, i = 0, 1, ¼, n-1,

starting with s₀ = s_j,0 = 1. The E_i = g(x_i)^-1, for i = 1, 2, ¼, n, are the corresponding location of errors in the vector w. It follows from [9, Theorem 7] that the denominator in Equation (3) is always a unit in .

Example 2.3. As in Example 2.1, if the received vector is given by b = (0, i, 0, 0), then the syndrome vector is given by s = bH^t = (ia⁵, ia², ia⁶). Applying the modified Berlekamp-Massey algorithm, we obtain the following table

Thus s⁽³⁾(z) = 1 + a⁴z. The root of r(z) = z + a⁴ (the reciprocal of s⁽³⁾(z)) is z₁ = a⁴. Among the elements 1, a, ¼ , a⁶ we have x₁ = a⁴ is such that x₁ - z₁ = 0 is a zero divisor in . Therefore, x₁ is the correct error-location number, and k₂ = 4 indicates that one error has occurred in the second coordinate of the codeword. The correct elementary symmetric function s₁ = a⁴ is obtained from x - x₁ = x - s₁ = x - a⁴. Finally, applying Forney's method to s and s₁, gives y₁ = i. Therefore, the error pattern is given by e = (0, i, 0, 0).

Example 2.4. As in Example 2.2, if the received vector is b = (0, 0, 1, 0, 0, 0, 0, 0, i, 0, 0), then the syndrome vector is given by

s = bH^t = (a¹² + ia¹⁴,a¹⁴ + ia⁸,a + ia²,a³ + ia¹¹)

Applying the modified Berlekamp-Massey algorithm, the following table isobtained

Thus s⁽⁴⁾(z) = 1 + a¹¹z + a¹¹z². The roots of r(z) = z² + a¹¹z + a¹¹ (the reciprocal of s⁽⁴⁾(z)) are z₁ = a² and z₂ = a⁹. Among the elements 1, a, a², ¼ , a¹⁴, we have x₁ = a² and x₂ = a⁹ are such that x₁ - z₁ = x₂ - z₂ = 0 are zero divisors in . Therefore, x₁ and x₂ are the correct error-location numbers and k₃ = 2 and k₉ = 9 indicates that two errors have occurred, one in position 3, and the other in position 9, in the codeword. The correct elementary symmetric functions s₁ and s₂ are obtained from (x-x₁)(x-x₂) = x²+s₁x+s₂. Thus, s₁ = s₂ = a¹¹. Finally, Forney's method applied to s, s₁ and s₂, gives s₁₁ = s₁ + x₁s₁₀ = a¹¹ + a² = a⁹ and s₂₁ = s₁ + x₂s₂₀ = a¹¹ + a⁹ = a². Thus, by Equation (3), we obtain y₁ = 1 and y₂ = i. Therefore, the error pattern is given by e = (0, 0, 1, 0, 0, 0, 0, 0, i, 0, 0).

3 Srivastava codes

In this section we define another subclass of alternant codes over local finite commutative rings which is very similar to the one proposed by J. N. Srivastava in 1967, in an unpublished paper [10], called Srivastava codes. These codes over finite fields are defined by parity-check matrices of the form

where a₁, a₂, ¼, a_r are distinct elements from GF(q^m) and b₁, b₂, ¼, b_n are all the elements in GF(q^m) except 0, and l > 0.

Definition 3.1. A shortened Srivastava code of length n < s over A has parity-check matrix

where a₁, ¼, a_n, b₁, ¼, b_r are n + m distinct elements of _s and l > 0.

Theorem 3.1. The Srivastava code has minimum Hamming distance d > r + 1.

Proof. The minimum Hamming distance of this code is at least r + 1 if and only if every combination of r or fewer columns of H is linearly independent over , or equivalently, that the submatrix

is nonsingular for any subset {i₁, ¼, i_r} of {1, 2, ¼, n}. The determinant of this matrix can be expressed as

where the matrix H₂ is given by

Note that det(H₂) is a Cauchy determinant of order r, and therefore we conclude that the determinant of the matrix H₁ is given by

where k = (-1)^m, m = and µ(x) = (x - b₁)(x - b₂) ¼ (x - b_r). Then by [9, Theorem 7] we have det(H₁) is a unit in and therefore d > r + 1.

Definition 3.2. Suppose r = kl and let a₁, ¼ , a_n, b₁, ¼ , b_k be n+k distinct elements of

_s, w₁, ¼ , w_n be elements of

_s. A generalized Srivastava code of length n < s over has parity-check matrix

where

for j = 1, 2, ¼ , k.

Theorem 3.2. The generalized Srivastava code has minimum Hamming distance d > kl + 1.

Proof. The proof of this theorem requires nothing more than the application of the Remark 2.1(3) and of the Theorem 3.1, since the matrices (1) and (9) are equivalent, with g(z) = (z - b_i)^l.

Example 3.1. As in Example 2.2, if

n = 8, r = 6, k = 2, l = 3,

{a₁, a₂, ¼, a₈} = {a⁴, a³, a⁵, a, a⁷, a¹², a¹⁰, a²},

{b₁, b₂} = {a⁹, a⁶} and {w₁, ¼, w₈} = {a, a, a², a⁴, a⁷, a¹⁰, a⁹, a³},

then the matrix

is the parity-check matrix of a generalized Srivastava code over

₂[i] of length 8 and minimum distance at least 7.

4 Conclusions

In this paper we presented construction and decoding procedure for Goppa codes over local finite commutative rings with identity. The decoding procedure is based on the modified Berlekamp-Massey algorithm. The complexity of the proposed decoding algorithm is essentially the same as that for Goppa codes over finite fields. Furthermore, we present the construction of Srivastava codes over local finite commutative rings with identity.

5 Acknowledgments

This paper has been supported by Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP - 02/07473-7.

Received: 3/VI/04. Accepted: 11/V/05

#609/04.

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Publication Dates

• Publication in this collection
  07 Nov 2005
• Date of issue
  Aug 2005

History

• Received
  03 June 2004
• Accepted
  11 May 2005