On Low Weight Codewords of Generalized Affine and Projective Reed-Muller Codes (Extended abstract)

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1 Designs, Codes and Cryptograpy manuscript No. (will be inserted by te editor) On Low Weigt Codewords of Generalized Affine and Projective Reed-Muller Codes (Extended abstract) Stépane Ballet Robert Rolland Received: date / Accepted: date Abstract We propose new results on low weigt codewords of affine and projective generalized Reed-Muller codes. In te affine case we give some results on codewords tat cannot reac te second weigt also called te next to minimal distance. In te projective case te second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of tis weigt are given. Keywords code codeword finite field generalized Reed-Muller code omogeneous polynomial yperplane ypersurface minimal distance next-to-minimal distance polynomial projective Reed-Muller code second weigt weigt Matematics Subject Classification (2010) 94B27 94B65 11G25 11T71 1 Introduction - Notations Tis paper proposes a study on low weigt codewords of generalized Reed-Muller codes and projective generalized Reed-Muller codes of degree d, defined over a finite field q, called respectively GRM codes and PGRM codes. For GRM codes, we give some results concerning te next to minimal weigt codewords. Tese codewords are known wen 1 d q 2 (cf. [10], [21]). For oter values of d we prove tat an irreducible, non-absolutely irreducible polynomial cannot reac te second weigt. For d we improve te previous result. More precisely we sow tat a polynomial aving a factor of degree d 2 wic is irreducible, non-absolutely irreducible, cannot reac te second weigt. For PGRM codes, we determine an upper bound and a lower bound for te second weigt of a PGRM code. Determining te low weigts of te Reed-Muller codes as well as te low weigt codewords are interesting questions related to various fields. Of course, from te point of view Stépane Ballet Aix-Marseille University, ERISCS and IML case 930, F13288 Marseille cedex 9, France stepane.ballet@univ-amu.fr Robert Rolland Aix-Marseille University, ERISCS and IML case 930, F13288 Marseille cedex 9 France robert.rolland@acrypta.fr

2 2 Stépane Ballet, Robert Rolland of coding teory, knowing someting on te weigt distribution of a code, and especially on te low weigts is a valuable information. From te point of view of algebraic geometry te problem is also related to te computation of te number of rational points of ypersurfaces and in particular ypersurfaces tat are arrangements of yperplanes. By means of incidence matrices, Reed-Muller codes are related to finite geometry codes (see [1, 5.3 and 5.4]). From tis point of view, te codewords ave a geometrical interpretation and can benefit from te numerous results in tis area. Consequently tere is a wide variety of concepts tat may be involved. 1.1 Polynomials and omogeneous polynomials Let q be te finite field wit q elements and n 1 an integer. We denote respectively by n qµ and È n qµ te affine space and te projective space of dimension n over q. Let q X 1 X 2 X n be te algebra of polynomials in n variables over q. If f is in q X 1 X 2 X n we denote by deg f µ its total degree and by deg Xi f µ its partial degree wit respect to te variable X i. Denote by qnµ te space of functions from n q into q. Any function in qnµ can be represented by a unique reduced polynomial f, namely suc tat for any variable X i te following olds: deg Xi f µ We denote by ÊP qnµ te set of reduced polynomials in n variables over q. Let d be a positive integer. We denote by ÊP qndµ te set of reduced polynomials P suc tat deg Pµ d. Remark tat if d n µ te set ÊP qndµ is te wole set ÊP qnµ. Let À qn 1dµ te space of omogeneous polynomials in n 1 variables over q wit total degree d. Te decomposition q X 0 X 1 X 2 X n Å d0 À qn 1dµ provides q X 0 X 1 X 2 X n wit a graded algebra structure. 1.2 Generalized Reed-Muller codes Let d be an integer suc tat 1 d n µ. Te generalized Reed-Muller code (GRM code) of order d over q is te following subspace of qn µ q : Ò Ó RM q dnµ f Xµ X ¾ n f ¾ q X 1 X n and deg f µ d q It may be remarked tat te polynomials f determining tis code are viewed as polynomial functions. Hence eac codeword is associated wit a unique reduced polynomial in ÊP qndµ. Let us denote by Z a f µ te set of zeros of f (were te index a stands for affine ). From a geometrical point of view Z a f µ is an affine algebraic ypersurface in n q and te number of points N a f µ #Z a f µ of tis ypersurface (te number of zeros of f ) is connected to te weigt W a f µ of te associated codeword by te following formula: W a f µ q n N a f µ

3 Low Weigt Codewords of Reed-Muller Codes 3 Te code RM q dnµ as te following parameters (cf. [12], [3, p. 72]) (were te index a stands for affine code ): 1. lengt m a qndµ q n, 2. dimension d n k a qndµ t0 j0 1µ j n t jq n 1 j t jq 3. minimum distance W 1µ a qndµ q bµq n a 1 were a and b are te quotient and te remainder in te Euclidean division of d by, namely d a µ b and 0 b. We denote by N 1µ a qndµ te maximum number of zeros for a non-null polynomial function of degree d were 1 d n µ, namely N 1µ a qndµ q n W 1µ a qndµ q n q bµq n a 1 Te minimum distance of RM q dnµ was given by T. Kasami, S. Lin, W. Peterson in [12]. Te words reacing tis bound were caracterized by P. Delsarte, J. Goetals and F. MacWilliams in [8]. 1.3 Projective generalized Reed-Muller codes Te case of projective codes is a bit different, because omogeneous polynomials do not define in a natural way functions on te projective space. Let d be an integer suc tat 1 d n µ. Te projective generalized Reed-Muller code of order d (PGRM code) was introduced by G. Lacaud in [14]. Let S a subset of n 1 q constituted by one point on eac punctured vector line of q n 1. Remark tat any point of te projective space È n qµ as a unique coordinate representation by an element of S. Te projective Reed-Muller code PRM q ndµ of order d over È n qµ is constituted by te words f Xµµ X ¾S were f ¾ À qn 1dµ and te null word: Ò Ó f Xµ X ¾S f ¾ À qn 1dµ 0 0µ PRM q ndµ Tis code is dependent on te set S cosen to represent te points of È n qµ. But te main parameters are independent of tis coice. Following [14] we can coose S n i0s i were S i 0 01X i 1 X n µ X k ¾ q. Subsequently, we sall adopt tis value of S to define te code PRM q ndµ. For a omogeneous polynomial f let us denote by Z f µ te set of zeros of f in te projective space È n qµ (were te index stands for projective ). From a geometrical point of view, an element f ¾ À qn 1dµ defines a projective ypersurface Z f µ in te projective space È n qµ. Te number N f µ #Z f µ of points of tis projective ypersurface is connected to te weigt W f µ of te corresponding codeword by te following relation: W f µ qn 1 1 N f µ Te parameters of PRM q ndµ are te following (cf. [23]) (were te index stands for projective code ):

4 4 Stépane Ballet, Robert Rolland 1. lengt m qndµ qn 1 1, 2. dimension k qndµ td mod 0tr n 1 j0 1µ j n 1 t j t jq n jq 3. minimum distance: W 1µ qndµ q bµq n a 1 were a and b are te quotient and te remainder in te Euclidean division of d 1 by, namely d 1 a µ b and 0 b. We denote by N 1µ qndµ te maximum number of zeros for a non-null omogeneous polynomial function of degree d were 1 d n µ, namely N 1µ qndµ qn 1 1 W 1µ qn 1 1 qndµ q bµq n a Minimal distance and corresponding codewords Te affine case: GRM codes For te affine case recall tat we write te degree d in te following form: d a µ b wit 0 b (1) Te minimum distance of a GRM code was given by T. Kasami, S. Lin, W. Peterson in [12]. Te words reacing tis bound (i.e. te polynomials reacing te maximal number of zeros) were caracterized by P. Delsarte, J. Goetals and F. MacWilliams in [8]. Suc a polynomial will be called a maximal polynomial and te associated ypersurface is called a maximal ypersurface. Te corresponding weigt is te minimal weigt Te projective case: PGRM codes Let us denote respectively by W 1µ qndµ and W 2µ qndµ te first and second weigt of te projective Reed-Muller code. In order to describe te minimal distance for te projective case, write d 1 a µ b wit 0 b. Te minimum distance of a PGRM code was given by J.-P. Serre for d q (cf. [22]), and by A. Sørensen in [23] for te general case. Te polynomials reacing te maximal number of zeros (or defining te minimum weigted codewords) are given by J.-P. Serre for d q (cf. [22]) and by te last autor (cf. [19]) for te general case. 2 Te second weigt in te affine case 2.1 wat is known Let us denote by W 2µ a qndµ te second weigt of te GRM code RM q dnµ, namely te weigt wic is just above te minimum distance. Several simple cases can be easily described. If d 1, we know tat te code as only tree weigts: 0, te minimum distance W 1µ a qn1µ q n q n 1 and te second weigt W 2µ a qn1µ q n. For d 2 and

5 Low Weigt Codewords of Reed-Muller Codes 5 q 2 te weigt distribution is more or less a consequence of te investigation of quadratic forms done by L. Dickson in [9] and was also done by E. Berlekamp and N. Sloane in an unpublised paper. For d 2 and any q (including q 2) te weigt distribution was given by R. McEliece in [18]. For q 2, for any n and any d, te weigt distribution is known in te range W 1µ a 2ndµ25W 1µ a 2ndµ by a result of Kasami, Tokura, Azumi [13]. In particular, te second weigt is W 2µ a 2ndµ 3 2 n d 1 if 1 d n 1 and W 2µ a 2ndµ 2 n d 1 if d n 1 or d 1. For d n µ te code RM q dnµ is trivial, namely it is te wole qdnµ, ence any integer 0 t q n is a weigt. Let us remark also tat if q p is a prime, GRM codes (and also PGRM codes) are te finite geometry codes, and in tis case te next to minimal distance is known as well as te geometrical nature of te corresponding codewords. Te general problem of te second weigt was tackled by D. Erickson in is tesis [10, 1974] and was partly solved. Unfortunately tis very good piece of work was not publised and remained virtually unknown. Meanwile several autors became interested in te problem. Te second weigt was first studied by J.-P. Cerdieu and R. Rolland in [7] wo proved tat wen q 2 is fixed, for d q sufficiently small te second weigt is W 2µ a qndµ q n dq n 1 d 1µq n 2 Teir result was improved by A. Sboui in [21], wo proved te formula for d q2. Te metods in [7] and [21] are of a geometric nature by means of wic te codewords reacing tis weigt were determined. Tese codewords are yperplane arrangements. Ten O. Geil in [11], using Gröbner basis metods, proved te formula for d q and solved te problem for n 2. Tis case is particularly important as we sall see later. Finally, te last autor in [20], using a mix of Geil s metod and geometrical considerations found te second weigt for all cases except wen d a µ 1. Recently, A. Bruen ([6]) exumed te work of Erickson and completed te proof, solving te problem of te second weigt for Generalized Reed-Muller code. Let us describe more precisely te result of Erickson. First, in order to present is result let us introduce te following notation used in [10]: s and t are integers suc tat d s µ t wit 0 t Teorem 1 Te second weigt W 2µ a qndµ is W 2µ a qndµ W 1µ a qndµ cq n s 2 were W 1µ a qndµ q tµq n s 1 is te minimal distance and c is q if s n 1 t 1 if s n 1 and 1 t q 1 2 or s n 1 and t 1 q if s 0 and t 1 c if q 4s n 2 and t 1 if q 3s n 2 and t 1 q if q 2s n 2 and t 1 q if q 40 s n 2 and t 1 c t if q 4s n 2 and q 1 2 t Te number c t is suc tat c t q tµq is te second weigt for te code RM q 2tµ.

6 6 Stépane Ballet, Robert Rolland Unfortunately te number c t is not determined in te work of Erickson. But, it results from te previous teorem tat if someone could calculate te second weigt for a case were c c t, te problem would be fully resolved. Alternatively, Erickson conjectured tat c t t 1 and reduced tis conjecture to a conjecture on blocking sets [10, Conjecture 4.14 p. 76]. Recently in [6] A. Bruen proved tat tis conjecture follows from two of is papers [4], [5]. Ten te problem is now solved by [10]+[6]. It is also solved by [10]+[11] (te important case n 2 is completely solved in [11] and tis leads to te conclusion as noted above) or by [10]+[20] (te cases not solved in [10] are explicitly resolved in [20]). Remark 2 Te values s and t are connected to te values a and b of te formula (1) in te following way: a s and b t unless t and in tis case a s 1 and b 0. Ten we can also express te second weigt wit te classical writing for te Euclidean quotient as in [20]. Finally let us remark tat we now ave several approaces, close to eac oter, but neverteless different. Te first one [10],[6] is mainly based on combinatorics of finite geometries, te second one [7],[21], [20] is mainly based on geometry and yperplane arrangements, te tird [11], [20] is mainly based on polynomial study by means of commutative algebra and Gröbner basis. All tese approaces can be fruitful for te study of similar problems, in particular for te similar codes based on incidence structures, finite geometry and incidence matrices (see [24], [16], [17], [15]). 2.2 New results on te codewords reacing te second weigt Te polynomials reacing te second weigt are known for 2d q (cf. [10, Teorem 3.13, p. 60], [21]). For te oter values of d te result is not known. However we can say tat: Teorem 3 If f ¾ ÊP qndµ is an irreducible polynomial but not absolutely irreducible, in n variables over q, of degree d 1 ten te weigt W a f µ of te corresponding codeword in RM q ndµ is suc tat W a f µ W 2µ a qndµ. Namely suc a polynomial cannot reac te next to minimal weigt. Teorem 4 If f ¾ ÊP qndµ is a product of two polynomials f g suc tat 1. 2 d ¼ deg gµ d deg f µ ; 2. g is irreducible but not absolutely irreducible; ten W a f µ W 2µ a qndµ. Namely suc a polynomial cannot reac te next to minimal weigt. Te proofs of tese two teorems will be given in te full paper and can be found in te preprint [2]. Remark 5 In any case, among te words reacing te second distance, tere are yperplane configurations. For example te yperplane configurations given in [20].

7 Low Weigt Codewords of Reed-Muller Codes 7 3 Te second weigt in te projective case In tis section we tackle te problem of finding te second weigt W 2µ qndµ for GPRM codes. Note tat if q is a prime p, Lemma 6 Let f be a omogeneous polynomial in n 1 variables of total degree d, wit coefficients in q, wic does not vanis on te wole projective space È n qµ. If tere exists a projective yperplane H suc tat te affine ypersurface È n qµ Ò Hµ Z f µ contains an affine yperplane of te affine space n qµ È n qµ Ò H ten te projective ypersurface Z f µ contains a projective yperplane. In particular if f restricted to te affine space n qµ defines a maximal affine ypersurface ten Z f µ contains a yperplane. Lemma 7 For n 2 te following olds W 1µ Proof Let us introduce te following notations: qn 1dµ W 2µ a qndµ W 2µ a qnd 1µ d 1 s d 1 µ t d 1 were 1 t d 1 ; d s d µ t d were 1 t d Te values c d 1µ and c dµ are te values of te coefficient c wic occurs in Teorem 1, wit respect to d 1 and d. Ten we ave Denote by te difference W 1µ qn 1dµ q t d 1 µq n s d 1 2 W 2µ a qndµ q t d µq n s d 1 c dµq n s d 2 W 2µ a qnd 1µ q t d 1 µq n s d 1 1 c d 1µq n s d 1 2 W 2µ a qnd 1µ W 1µ qn 1dµ W 2µ a qndµ If 1 t d 1 q 2 ten q 2, t d t d 1 1 and s d s d 1. In tis case let us denote by s te common value of s d and s d 1. Hence q n s 2 t d 1 c d 1µ c dµ If s n 1 ten c d 1µ c dµ q and 0. If s n 1 and 1 t d 1 q ten q 4 and c d 1µ c dµ 1. Hence 0. If s n 1 and q t d 1 q 1 2 ten q 4 and c d 1µ c dµ 1. Hence 0. If s n 1, q 4 and t d 1 1 ten c d 1µ c dµ q t d 1. Hence 0. If s n 1 q 3 and t d 1 1 ten c d 1µ c dµ 1. Hence 0. If t d 1 ten t d 1 and s d s d 1 1. Hence q n s d 1 3 c d 1µq c dµ 0

8 8 Stépane Ballet, Robert Rolland Teorem 8 Let W 2µ qndµ be te second weigt for a omogeneous polynomial f in n 1 variables (n 2) of total degree d, wit coefficients in q, wic is not maximal. Let us define V 2µ qndµ by: V 2µ qndµ 2 if d n µ and V 2µ Ten te following olds qndµ W 1µ qn 1dµ W 2µ V 2µ Proof Let us remark first tat by Lemma 7 qndµ W 2µ qndµ W 2µ a qnd 1µ V 2µ qndµ W 2µ a qnd 1µ a qndµ if d n µ (2) If d n µ, as f does not vanis on te wole projective space È n qµ, and f is not maximal ten N f µ qn Tis bound is attained. Ten in tis case W 2µ qndµ 2. Suppose now tat 2 d n µ. Let f suc tat Z f µ is not maximal. Suppose first tat tere is an yperplane H in Z f µ. Ten we can suppose tat f X 0 X 1 X n µ X 0 g X 0 X 1 X n µ were g is an omogeneous polynomial of degree d 1. Te function f 1 X 1 X n µ g 1X 1 X n µ defined on te affine space n qµ È n qµ Ò H is a polynomial function in n variables of total degree d 1. If it was maximum, by [19, Lemma 2.3] te function f would also be maximum. Ten #Z a f 1 µ q n W 2µ a qnd 1µ. Hence te following olds: #Z f µ qn 1 qn W 2µ a qnd 1µ #Z f µ qn 1 1 W 2µ a qnd 1µ and te equality olds if and only if f 1 reaces te second weigt on te affine space n qµ. Tis case actually occurs. Hence for suc a word, in general we ave W f µ W 2µ a qnd 1µ and as te equality occurs, te following olds for te second distance: W 2µ qndµ W 2µ a qnd 1µ Suppose now tat tere is not any yperplane in te ypersurface Z f µ. Let H be a yperplane and n qµ È n qµ Ò H. Ten as H Z f µ H # H Z f µµ qn 1 W 1µ qn 1dµ

9 Low Weigt Codewords of Reed-Muller Codes 9 and by Lemma 6 Ten and consequently # Z f µ n qµ q n W 2µ #Z f µ qn 1 qn 1 1 W f µ W 1µ W 1µ a qndµ qn 1dµ q n W 2µ a qndµ W 1µ qn 1dµ W 2µ a qndµ qn 1dµ W 2µ a qndµ Ten, for te second distance te conclusion of te teorem olds. References 1. Assmus, E., Key, J.: Designs and teir Codes, Cambridge Tracts in Matematics, vol Cambridge University Press (1992) 2. Ballet, S., Rolland, R.: Remarks on Low Weigt Codewords of Generalized Affine and Projective Reed- Muller Codes. arxiv; v3 (2012) 3. Blake, I., Mullin, R.: Te Matematical Teory of Coding. Academic Press (1975) 4. Bruen, A.: Polynomial Multiplicities over Finite Fields and Intersection Sets. Journal of Combinatorial Teory 60(1), (1992) 5. Bruen, A.: Applications of Finite Fields to Combinatorics and Finite Geometries. Acta Applicandae Matematicae 93(1 3) (2006) 6. Bruen, A.: Blocking Sets and Low-Weigt Codewords in te Generalized Reed-Muller Codes. In: A. Bruen, D. Welau, C.M. Society (eds.) Error-correcting Codes, Finite Geometries, and Cryptograpy, Contemporary Matematics, vol. 525, pp American Matematical Society (2010) 7. Cerdieu, J.P., Rolland, R.: On te Number of Points of Some Hypersurfaces in n q. Finite Field and teir Applications 2, (1996) 8. Delsarte, P., Goetals, J., MacWilliams, F.: On Generalized Reed-Muller Codes and teir Relatives. Information and Control 16, (1970) 9. Dickson, L.: Linear Groups. Dover Publications (1958) 10. Erickson, D.: Counting Zeros of Polynomials over Finite Fields. P.D. tesis, Tesis of te California Institute of Tecnology, Pasadena California (1974) 11. Geil, O.: On te Second Weigt of Generalized Reed-Muller codes. Designs,Codes and Cryptograpy 48(3), (2008) 12. Kasami, T., Lin, S., Peterson, W.: New Generalizations of te Reed-Muller Codes Part I: Primitive Codes. IEEE Transactions on Information Teory IT-14(2), (1968) 13. Kasami, T., Tokura, N., Azumi, S.: On te Weigt Enumeration of Weigts less tan 25d of Reed-Muller Codes. Information and Control 30(4), (1976) 14. Lacaud, G.: Projective Reed-Muller Codes. In: Coding Teory and Applications, no. 311 in Lecture Notes in Computer Science, pp Springer-Verlag (1988) 15. Lavrauw, M., Storme, L., Sziklai, P., Van de Voorde, G.: An Empty Interval in te Spectrum of Small Weigt Codewords in te Code from Points and k-spaces in PG nqµ. Journal of Combinatorial Teory 16. Lavrauw, M., Storme, L., Van de Voorde, G.: On te Code Generated by te Incidence Matrix of Points and Hyperplanes in PG nqµ and its Dual. Designs,Codes and Cryptograpy 48, (2008) 17. Lavrauw, M., Storme, L., Van de Voorde, G.: On te Code Generated by te Incidence Matrix of Points and k-spaces in PG nqµ and its Dual 14, (2008) 18. McEliece, R.: Quadratic Forms over Finite Fields and Second-Order Reed-Muller Codes. Tec. rep., JPL Space Programs Summary III (1969) 19. Rolland, R.: Number of Points of Non-Absolutely Irreducible Hypersurfaces. In: Algebraic Geometry and its Applications, Number Teory and Its Applications, vol. 5, pp World Scientific (2008). Proceedings of te first SAGA Conference, 7-11 May 2007, Papeete 20. Rolland, R.: Te Second Weigt of Generalized Reed-Muller Codes in Most Cases. Cryptograpy and Communications Discrete Structures, Boolean Functions and Sequences 2(1), (2010)

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