In this paper, we describe a procedure for constructing $q$--ary
$[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for
$q$ even), using a set of non--degenerate Hermitian forms in $PG(2,q^2)$.Comment: 8 Pages Minor typesetting changes; dedicatio

Aguglia, A.

Giuzzi, L.

English

arXiv

In this paper, we describe a procedure for constructing $q$--ary
$[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or
$N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian
forms in $PG(2,q^2)$

AGUGLIA A

GIUZZI, Luca

Archivio istituzionale della ricerca - Università di Brescia

Construction of a 3-Dimensional MDS code

In this paper, we describe a procedure for constructing $q$--ary

$[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or

$N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian

forms in $PG(2,q^2)$

Aguglia, Angela

Giuzzi, Luca

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 3, Number 1, Pages 39–46ISSN 1715-0868CONSTRUCTION OF A 3-DIMENSIONAL MDS-CODEANGELA AGUGLIA AND LUCA GIUZZIDedicated to the centenary of the birth of Ferenc Ka´rteszi (1907–1989).Abstract. In this paper, we describe a procedure for constructing q–ary [N, 3, N−2]–MDS codes, of length N ≤ q+1 (for q odd) or N ≤ q+2(for q even), using a set of non–degenerate Hermitian forms in PG(2, q2).1. IntroductionThe well–known Singleton bound states that the cardinality M of a codeof length N with minimum distance d over a q–ary alphabet always satisfies(1.1) M ≤ qN−d+1;see [8]. Codes attaining the bound are called maximum distance separablecodes, or MDS codes for short.Interesting families of maximum distance separable codes arise from geo-metric and combinatorial objects embedded in finite projective spaces. Inparticular linear [N, k,N − k + 1]–MDS codes, with k ≥ 3, and N–arcs inPG(k − 1, q) are equivalent objects; see [1].A general method for constructing a q–ary code is to take N polynomialsf1, . . . , fN in n indeterminates, defined over GF(q), and consider the set Cgiven byC = {(f1(x), . . . , fN (x)) | x ∈ W},where W is a suitable subset of GF(q)n. In this paper, we deal with the case|W| = qt and also assume that the evaluation functionΘ : W → Cx 7→ (f1(x), f2(x), . . . , fN (x))is injective.If C attains the Singleton bound then the restrictions of all the codewordsto any given t = N − d + 1 places must all be different, namely in any tpositions all possible vectors occur exactly once. This means that a necessarycondition for C to be MDS is that any t of the varieties V (fm) for m =1, . . . , N meet in exactly one point in W. Here V (f) denotes the algebraicvariety associated to f .2000 Mathematics Subject Classification. 05B25, 11E39.Key words and phrases. Hermitian form, MDS-code.c©2008 University of Calgary3940 ANGELA AGUGLIA AND LUCA GIUZZIApplying the above procedure to a set of non–degenerate Hermitian formsin PG(2, q2) we construct some q–ary [N, 3, N − 2]–MDS codes, of lengthN ≤ q + 1 (for q odd) or N ≤ q + 2 (for q even). The codes thus obtainedcan also be represented by sets of points in PG(3, q); this representation isused in Section 4 in order to devise an algebraic decoding procedure, basedupon polynomial factorisation; see [10].2. PreliminariesLet A be a set containing q elements. For any integer N ≥ 1, the functiondH : AN ×AN 7→ N given bydH(x,y) = |{i : xi 6= yi}|,is a metric on AN . This function is called the Hamming distance on AN . Aq–ary (N,M, d)–code C over the alphabetA is just a collection of M elementsof AN such that any two of them are either the same or at Hamming distanceat least d; see [4, 6]. The elements of C are called codewords whereas theintegers d and N are respectively the minimum distance and the length ofC.IfA = GF (q) and C is a k–dimensional vector subspace of GF (q)N , then Cis said to be a linear [N, k, d]–code. Under several communication models,it is assumed that a received word r should be decoded as the codewordc ∈ C which is nearest to r according to the Hamming distance; this isthe so–called maximum likelihood decoding. Under these assumptions thefollowing theorem, see [4, 6], provides a basic bound on the guaranteed errorcorrection capability of a code.Theorem 2.1. If C is a code of minimum distance d, then C can alwayseither detect up to d− 1 errors or correct e = b(d− 1)/2c errors.Observe that the theorem does not state that it is not possible to decodea word when more than e errors happened, but just that in this case thecorrection may fail. Managing to recover from more than e errors for somegiven received codewords is called “correcting beyond the bound”.The weight of an element x ∈ GF (q)N is the number of non–zero compo-nents xi of x. For a linear code the minimum distance d equals the minimumweight of the non–zero codewords.The parameters of a code are not independent; in general it is difficult todetermine the maximum number of words a code of prescribed length N andminimum distance d may contain. For any arbitrary linear [N, k, d]–code,condition (1.1) may be rewritten as(2.1) d ≤ N − k + 1;thus C is a linear MDS code if and only if equality holds in (2.1).In Section 3 we shall make extensive use of some non–degenerate Hermit-ian forms in PG(2, q2).CONSTRUCTION OF A 3-DIMENSIONAL MDS-CODE 41Consider the projective space PG(d, q2) and let V be the underlying vec-tor space of dimension d + 1. A sesquilinear Hermitian form is a maph : V × V −→ GF (q2)additive in both components and satisfyingh(kv, lw) = klqh(v,w)for all v,w ∈ V and k, l ∈ GF (q2). The form is degenerate if and only if thesubspace {v | h(v,w) = 0 ∀w ∈ V }, the radical of h, is different from {0}.Given a sesquilinear Hermitian form h, the associated Hermitian variety His the set of all points of PG(d, q2) such that {〈v〉 | 0 6= v ∈ V, h(v,v) = 0}.The variety H is degenerate if h is degenerate; non–degenerate otherwise.If h is a sesquilinear Hermitian form in PG(d, q2) then the map F : V −→GF (q) defined byF (v) = h(v,v),is called the Hermitian form on V associated to h. The Hermitian form F isnon–degenerate if and only if h is non–degenerate. Complete introductionsto Hermitian forms over finite fields may be found in [2, 7].3. ConstructionLet S be a representative system for the cosets of the additive subgroupT0 of GF(q2) given byT0 = {y ∈ GF(q2) : T (y) = 0},whereT : GF(q2) → GF(q)y 7→ yq + yis the trace function. Denote by Λ a subset of GF (q2) satisfying(3.1)(α− βγ − β)q−16= 1for any α, β, γ ∈ Λ. Choose a basis B = {1, ε} of GF (q2), regarded as a2–dimensional vector space over GF (q); hence, it is possible to write eachelement α ∈ GF (q2) in components α1, α2 ∈ GF(q) with respect to B. Wemay thus identify the elements of GF (q2) with the points of AG(2, q), bythe bijection which maps (x, y) ∈ AG(2, q) to x + εy ∈ GF (q2). Condition(3.1) corresponds to require that Λ, regarded as point–set in AG(2, q), is anarc. Thus, setting N = |Λ|, we have(3.2) N ≤{q + 1 for q odd,q + 2 for q even;see [5, Theorem 8.5].Now, consider the non–degenerate Hermitian formsFλ(X,Y,Z) on GF (q2)3Fλ(X,Y,Z) = Xq+1 + Y qZ + Y Zq + λqXqZ + λXZq,42 ANGELA AGUGLIA AND LUCA GIUZZIas λ varies in Λ. Label the elements of Λ as λ1, . . . , λN and let Ω = GF (q2)×S.Theorem 3.1. The setC = {(Fλ1(x, y, 1),Fλ2 (x, y, 1), . . . ,FλN (x, y, 1))|(x, y) ∈ Ω}is a q-ary linear [N, 3, N − 2]–MDS code.Proof. We first show that C consists of q3 tuples from GF (q). Let (x0, y0), (x1, y1) ∈Ω and suppose that for any λ ∈ Λ,Fλ(x0, y0, 1) = Fλ(x1, y1, 1).Then,(3.3) T (λ(x1 − x0)) = xq+10 − xq+11 + T(y0 − y1).In particular,(3.4) T (λ(x1 − x0)) = T (α(x1 − x0)) = T (γ(x1 − x0))for any α, λ, γ ∈ Λ.If it were x1 6= x0, then (3.4) would imply(α− βγ − β)q−1= 1,contradicting the assumption made on Λ. Therefore, x1 = x0 and from (3.3)we get T (y0 − y1) = 0. Hence, y0 and y1 are in the same coset of T0; bydefinition of S, it follows that y0 = y1, thus C has as many tuples as |Ω|.We are now going to show that C is a vector subspace of GF(q)N . Take(x0, y0), (x1, y1) ∈ Ω. For any λ ∈ Λ,(3.5) Fλ(x0, y0, 1) + Fλ(x1, y1, 1) = Fλ(x2, y2, 1),where x2 = x0 + x1 and y2 = y0 + y1 − xq0x1 − xq1x0. Likewise, for anyκ ∈ GF(q),(3.6) κFλ(x0, y0, 1) = Fλ(x, y, 1),where x = κx0 and y is a root ofy2 + y = (κ− κ2)xq+10 + κ(yq0 + y0).Therefore, C is a vector subspace of GF(q)N ; as it consists of q3 tuples, C isindeed a 3–dimensional vector space.Finally we prove that the minimum distance d of C is N − 2. Since C isa vector subspace of GF(q)N , its minimum distance is N − z, wherez = maxc∈C,c6=0|{i : ci = 0}|.CONSTRUCTION OF A 3-DIMENSIONAL MDS-CODE 43Furthermore, z ≥ 2 because of Singleton bound (2.1). In order to show thatz = 2 we study the following system(3.7)Fα(x, y, 1) = 0,Fβ(x, y, 1) = 0,Fγ(x, y, 1) = 0,for α, β, γ distinct elements of Λ. Set U = xq+1+yq +y, V = xq and W = x;then, (3.7) becomes(3.8)U + αqV + αW = 0,U + βqV + βW = 0,U + γqV + γW = 0.Since(α−βγ−β)6= 1, the only solution of (3.8) is U = V = W = 0, that isx = 0 and y + yq = 0. In particular, there is just one solution to (3.7) in Ω,that is x = (0, 0). This implies that a codeword which has at least three zerocomponents is the zero vector, hence z = 2 and thus the minimum distanceof C is N − 2. Example 3.2. When q is odd, a representative system S for the cosets ofT0 is given by the subfield GF(q) embedded in GF(q2). In this case it isthen extremely simple to construct the code. For q = 5, a computation usingGAP [3], shows that in order for Λ to satisfy property (3.1), we may takeΛ = {ε3, ε4, ε8, ε15, ε16, ε20}, where ε is a root of the polynomial X2−X +2,irreducible over GF (5). The corresponding Hermitian forms areXq+1 + Y qZ + yZq + ε15XqZ + ε3XZq,Xq+1 + Y qZ + Y Zq + ε20XqZ + ε4XZq,Xq+1 + Y qZ + Y Zq + ε16XqZ + ε8XZq,Xq+1 + Y qZ + Y Zq + ε3XqZ + ε15XZq,Xq+1 + Y qZ + Y Zq + ε8XqZ + ε16XZq,Xq+1 + Y qZ + Y Zq + ε4XqZ + ε20XZq.A generator matrix for the [6, 3, 4]–MDS code obtained applying Theorem3.1 to these Hermitian forms isG =1 1 1 1 1 10 1 0 2 1 20 0 1 2 2 1Remark 3.3. In PG(2, q2), take the line `∞ : Z = 0 as the line at infinity.Then, in the affine plane AG(2, q2) = PG(2, q2) \ `∞, any two Hermitiancurves V (Fλ) have q2 affine points in common, q of which in Ω ⊂ AG(2, q2).Likewise, the full intersection ⋂λ∈ΛV (Fλ)44 ANGELA AGUGLIA AND LUCA GIUZZIconsists of the q affine points {(0, y) | yq + y = 0}, corresponding to just asingle point in Ω.Remark 3.4. Denote by Ai the number of words in C of weight i. Since C isan MDS code, we haveAi =(Ni)(q − 1)i−N+2∑j=0(−1)j(i− 1j)qi−j−N+2;see [9]. Thus,AN−2 =12(N2 −N)(q − 1),AN−1 = Nq2 − (N2 −N)q + N 2 − 2N,AN = q3 −Nq2 +12((N2 −N)q −N 2 + 3N).4. DecodingIn this section it will be shown how the code C we constructed may bedecoded by geometric means.Our approach is based upon two remarks:(1) Any received word r = (r1, . . . , rN ) can be uniquely represented bya set r˜ of N points of PG(3, q)r˜ ={(λ1i , λ2i , ri, 1): λ = λ1i + ελ2i ∈ Λ}.These points all lie on the cone Ψ of basisΞ ={(λ1i , λ2i , 0, 1): λ = λ1i + ελ2i ∈ Λ}and vertex Z∞ = (0, 0, 1, 0).(2) For any a, b ∈ GF(q2), the functionφ(a,b)(x, y, z, t) =(aq+1 + T(b))t + T((x + εy)a)is a homogeneous linear form with domain GF(q)4.Recall that the codeword c corresponding to a given (a, b) ∈ Ω isc =(φ(a,b)(λ11, λ21, 0, 1), φ(a,b)(λ12, λ22, 0, 1), . . . , φ(a,b)(λ1N , λ2N , 0, 1));thus, c˜, the set containing the points (λ1i , λ2i , ci, 1), is the full intersection ofthe plane pia,b : z = φ(a,b)(x, y, z, t) with the cone Ψ.It is clear that knowledge of the plane pi(a,b) is enough to reconstructthe codeword c. In the presence of errors, we are looking for the nearestcodeword c to a vector r; this is the same as to determine the plane pi(a,b)containing most of the points of r˜. In order to obtain such a plane, weadopt the following approach. Assume ` to be a line of the plane pi0,0 : z = 0external to Ξ and denote by pi∞ the plane at infinity of equation t = 0. Forany P ∈ `, let r˜P be the projection from P of the set r˜ on pi∞. Write LPrfor acurve of pi∞ of minimum degree containing r˜P . Observe that degLPr≤ q+1and degLPr = 1 if, and only if, all the points of r˜ lie on a same plane throughCONSTRUCTION OF A 3-DIMENSIONAL MDS-CODE 45P , that is r˜ corresponds to a codeword associated with that plane passingthrough P .We now can apply the following algorithm using, for example, [3].(1) Take P ∈ `;(2) Determine the projection rP and compute the curve LPr;(3) Factor LPrinto irreducible factors, say L1,L2, . . . ,Lv;(4) Count the number of points in r˜P ∩ V (Li) for any factor Li of LPrwith degLi = 1;(5) If for some i we have ni > (N +1)/2, then return the plane spawnedby P and two points of Li; else, as long as not all the points of `have been considered, return to point 1;(6) If no curve with the required property has been found, return failure.Remark 4.1. The condition on ni in point (5) checks if the plane containsmore than half of the points corresponding to the received word r; when thisis the case, a putative codeword c is constructed, with d(c, r) ≤ (N − 3)/2;thus, when c ∈ C, then it is indeed the unique word of C at minimum distancefrom r. However, the aforementioned algorithm may be altered in severalways, in order to be able to try to correct errors beyond the bound; possibleapproaches are:(1) iterate the procedure for all the points on ` and return the planescontaining most of the points corresponding to the received vector;(2) use some further properties of the cone Ψ; in particular, when Ξ is aconic it seems possible to improve the decoding by considering alsothe quadratic components of the curve LPr.Remark 4.2. The choice of P on a line ` is due to the fact that any line ofpi0,0 meets all the planes of PG(3, q). In general, we might have chosen ` tobe just a blocking set disjoint from Ξ. If q is odd and |Λ| = q + 1, then theline ` is just an external line to a conic of pi0,0.References1. E. F. Assmus, Jr. and J. D. Key, Designs and their codes, Cambridge University Press,1992.2. S. N. Bose and I. M. Chakravarti, Hermitian varieties in a finite projective spacePG(N, q2), Canad J. Math. 18 (1966), 1161–1182.3. GAP – Groups, Algorithms, and Programming, Version 4.4, 2006, http://www.gap-system.org.4. L. Giuzzi, Codici correttori, Springer-Verlag, 2006.5. J. W. P. Hirschfeld, Projective geometries over finite fields, 2 ed., Oxford MathematicalMonographs, The Clarendon Press, Oxford University Press, New York, 1998.6. R. McEliece, Theory of information and coding, Cambridge University Press, 2002.7. B. Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finito, Ann.Mat. Pura Appl. (4) 70 (1965), 1–201.8. R. C. Singleton, Maximum distance q–nary codes, IEEE Trans. Inf. Theory 10 (1964),no. 2, 116–118.9. L. Staiger, On the weight distribution of linear codes having dual distance d′ > k,IEEE Trans. Inf. Theory 35 (1989), no. 1, 186–188.46 ANGELA AGUGLIA AND LUCA GIUZZI10. M. Sudan, Decoding Reed Solomon codes beyond the error-correction bound, Journalof Complexity 13 (1997), no. 1, 180–193.Dipartimento di Matematica, Politecnico di Bari,Via G. Amendola 126/B, 70126 Bari, ItalyE-mail address: a.aguglia@poliba.itE-mail address: l.giuzzi@poliba.it

Construction of a 3-Dimensional MDS code

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