Department of Mathematcal Scences, Clemson Unversty http://www.ces.clemson.edu/ keyj/ The mnmum weght of dual codes from projectve planes J. D. Key keyj@ces.clemson.edu 1/21 P
Abstract The mnmum weght and the nature of the mnmum-weght vectors of the p-ary codes from projectve planes of order dvsble by p was establshed n the 1960s, at an early stage of the study of these codes. The same cannot be sad for the duals of these codes, where, n general, nether the mnmum weght nor the nature of the mnmum-weght words s known. Ths talk wll provde a survey of what s known of ths problem, what progress has been made recently, and gve some new bounds for planes of some specfc orders. October 8, 2006 2/21 P
Codes from planes Some old results from the folklore, taken from [AK92]: Theorem 1 Let Π be a projectve plane of order n and let p be a prme dvdng n. The mnmum-weght vectors of C p (Π), are precsely the scalar multples of the ncdence vectors of the lnes,.e. av L, where a F p, a 0, and L s a lne of Π. The mnmum weght of C p (Π) s at least n + 2. If the mnmum weght s n + 2 then, p = 2 and n s even, n whch case the mnmum-weght vectors are all of the form v X where X s a hyperoval of Π. Theorem 2 If π s an affne plane of order n and p s a prme dvdng n, then the mnmum weght of C p (π) s n and all mnmum-weght vectors are constant. If n = p the mnmum-weght vectors of C p (π) are precsely the scalar multples of the ncdence vectors of the lnes of π. 3/21 P
Desarguesan planes Theorem 3 Let p be any prme, q = p t, and Π = P G 2 (F q ). Then C p (Π) has dmenson ( ) p+1 t 2 + 1. The mnmum-weght vectors of Cp (Π) are the scalar multples of the ncdence vectors of the lnes. The mnmum weght d of C p (Π) satsfes q + p d 2q, wth equalty at the lower bound f p = 2. If π = AG 2 (F q ), then C p (π) has dmenson ( ) p+1 t. 2 The mnmum-weght vectors of C p (π) are the scalar multples of the ncdence vectors of the lnes of π. The mnmum weght d of C p (π) satsfes q + p d 2q, wth equalty at the lower bound when p = 2. 4/21 P
Bnary codes If a projectve plane of even order n does not have hyperovals, the next possble weght n C 2 (Π) s n + 4. A non-empty set S of ponts n a plane s of even type f every lne of the plane meets t evenly. Then S and the order n of the plane must be even, and that S = n + 2s, where s 1. A set of ponts has type (n 1, n 2,..., n k ) f any lne meets t n n ponts for some, and for each there s at least one lne that meets t n n ponts. So the set s of even type f all the n are even. If a set S of sze n + 4 n a plane of even order n s of even type, then t s of type (0, 2, 4). 5/21 P
Korchmáros and Mazzocca [KM90] consder (n + t)-sets of type (0, 2, t) n the desarguesan plane of order n. They show that sets of sze n + 4 that are of type (0, 2, 4) always exst n the desarguesan plane for n = 4, 8, 16, but have no exstence results for sze n + 4 for n > 16. From Key and de Resmn [KdR98]: Theorem 4 Let Π be any of the known planes of order 16. Then Π has a 20-set of even type. (Two of these planes do not have hyperovals.) Incorrect exercse from [AK92, page 214]: If Π = P G 2 (F 2 m), where m 3 and C = C 2 (Π), show that f c C satsfes wt(c) > 2 m + 2, then wt(c) 2 m + 8. (Probably true for m 5.) Blokhus, Szőny and Wener [BSW03], Gács and Wener [GW03], and Lmbupasrporn [Lm05], further explore sets of even type. 6/21 P
Odd-order planes The mnmum weght of the dual code of planes of odd order s only known n general for desarguesan planes of prme order p (when t s 2p), and for some planes of small order. The followng results appeared n Clark and Key [CK99], and part of them much earler n Sachar [Sac79]: Theorem 5 If D s a projectve plane of odd order q = p t, then 1. d 4 3 q + 2; 2. f p 5 then d 3 2 q + 2. (Ths s better than the bound p + q for desarguesan planes.) Theorem 6 A projectve plane of square order q 2 that contans a Baer subplane has words of weght 2q 2 q n ts p-ary dual code, where p q. 7/21 P
Translaton planes From Clark, Key and de Resmn [CKdR02]: Theorem 7 Let Π be a projectve translaton plane of order q m (e.g. P G 2 (F m q )) where m = 2 or 3, q = p t, and p s a prme. If C s ts p-ary code then C has words of weght 2q m (q m 1 + q m 2 + + q). If Π s desarguesan, ths also holds for m = 4. We really want ths for all m 2 to get an upper bound for the mnmum weght of C better than 2q m. But, we couldn t verfy our constructon for m 5. For the desarguesan plane of order p m, where p s a prme, n all cases where the mnmum weght of the dual p-ary code s known, and n partcular for p = 2, or for m = 1, the mnmum weght s precsely as gven n ths formula, 2p m (p m 1 + p m 2 + + p). 8/21 P
Queston 1 Is the mnmum weght of the dual code of the p-ary code of the desarguesan plane of order p m gven by the formula for all prmes p and all m 1 2p m (p m 1 + p m 2 + + p) = 2p m + 1 pm 1 p 1 9/21 P
Fgueroa planes A smlar constructon as that used n Theorem 7 apples to Fgueroa planes: see Key and de Resmn [KdR03]. Theorem 8 Let Φ be the Fgueroa plane Fg(q 3 ) of order q 3 where q = p t and p s any prme. Let C denote the p-ary code of Φ. Then C contans words of weght 2q 3 q 2 q. Furthermore, f d denotes the mnmum weght of C then 1. d = q + 2 f p = 2; 2. 4 3 q + 2 d 2q 3 q 2 q f p = 3; 3. 3 2 q + 2 d 2q 3 q 2 q f p > 3. 10/21 P
Planes of order 9 The other odd orders for whch the mnmum weght s known n the desarguesan case are q = 9 (see [KdR01]) and q = 25 (see [Cla00, CHKW03]). From Key and de Resmn [KdR01]: Theorem 9 Let Π be a projectve plane of order 9. The mnmum weght of the dual ternary code of Π s 15 f Π s Φ, Ω, or Ω D, and 14 f Π s Ψ. The four projectve planes of order 9 are: the desarguesan plane, Φ, the translaton (Hall) plane, Ω, the dual translaton plane, Ω D, and the Hughes plane, Ψ. The weght- 15 vectors are from the Baer subplane constructon; the weght-14 are from two totally dsjont (share no ponts nor lnes) Fano planes. 11/21 P
Planes of order 25 From Clark, Hatfeld, Key and Ward [CHKW03]: Theorem 10 If Π s a projectve plane of order 25 and C s the code of Π over F 5, then the mnmum weght d of C s ether 42 or 44, or 45 d 50. If Π has a Baer subplane, then the mnmum weght s ether 42, 44 or 45. If the mnmum weght s 42, then a mnmum-weght word has support that s the unon of two projectve planes, π 1 and π 2, of order 4 that are totally dsjont (share no ponts nor lnes) and the word has the form v π 1 v π 2. If the mnmum weght s 44 then the support of a mnmum-weght word s the unon of two dsjont complete 22-arcs that have eleven 2-secants n common. If the mnmum weght s 45 then v β v l, where β s a Baer subplane of Π and l s a lne of Π that s a lne of the subplane, s a mnmum-weght word. 12/21 P
Corollary 11 The dual 5-ary code of the desarguesan projectve plane P G 2 (F 25 ) has mnmum weght 45. All the known planes of order 25 have Baer subplanes. Czerwnsk and Oakden [CO92] found the 21 translaton planes of order 25. 13/21 P
Planes of order 49 Work for masters project of Fdele Ngwane [KN] at Clemson: Theorem 12 If C s the 7-ary code of a projectve plane of order 49, then the mnmum weght of the dual code C s at least 88. Thus, the mnmum weght d of C satsfes 88 d 98. Further, 88 d 91 f the projectve plane contans a Baer subplane. Note that a word of weght 86 that conssts of two totally dsjont 2-(43,6,1) desgns (.e. planes of order 6), a combnatoral possblty, cannot exst by Bruck-Ryser. Mathon and Royle [MR95] fnd that there are 1347 translaton planes of order 49. 14/21 P
Totally dsjont sets Let Π be a projectve plane of order n, and let p n, where p s a prme. Let S for {1, 2} be a set of s ponts of Π that s a (0, 1, h )-set, where h > 1,.e. lnes meet S n 0, 1 or h ponts. S 1 and S 2 are totally dsjont f for {, j} = {1, 2}, they have no ponts n common; the h -secants to S are exteror to S j ; every 1-secant to S s a 1-secant to S j. Proposton 13 If Π s a projectve plane of order n, p a prme dvdng n, and S for = 1, 2 are a par of totally dsjont (0, 1, h )-sets, respectvely, where p h, then v S 1 v S 2 s a word of weght s 1 + s 2 n C p (Π) where S = s. Further n + 1 = s 1 1 h 1 1 + s 2 = s 2 1 h 2 1 + s 1. 15/21 P
The followng specal cases are feasble: 1. s 1 = s 2 = h 1 = h 2 : the confguraton conssts of two lnes wth the pont of ntersecton omtted. 2. If n = q r = p t and s 2 = h 2, then p h 1 and (h 1 1) (q r 1) s possble f h 1 = q, whch wll gve a word of weght 2q r (q r 1 + q r 2 +... + q). Ths s the constructon of our Theorem 7. 16/21 P
Other numercal possbltes 1. n = 9, s 1 = s 2 = 7, h 1 = h 2 = 3: two totally dsjont Fano planes, weght 14 (see [KdR01]). 2. n = 25, s 1 = s 2 = 21, h 1 = h 2 = 5: two totally dsjont planes of order 4, weght 42; n general t s unknown f a plane of order 25 can have an embedded plane of order 4 (see the note below). 3. n = 27, s 1 = s 2 = 19, h 1 = h 2 = 3: two totally dsjont Stener trple systems, weght 38; t s not known f ths s possble. 4. n = 27, s 1 = 25, s 2 = 16, h 1 = 3, h 2 = 6: 2-(25, 3, 1) and 2-(16, 6, 1) desgns, weght 41; no desgn wth the latter parameters can exst by Fsher s nequalty. 5. n = 49, s 1 = s 2 = 43, h 1 = h 2 = 7: two totally dsjont 2-(43,7,1) desgns,.e. planes of order 6, weght 86; planes of order 6 do not exst, by the Bruck-Ryser theorem (see, for example, [AK92, Chapter 4]). 6. n = 81, s 1 = 73, s 2 = 46, h 1 = 3, h 2 = 6: 2-(73, 3, 1) and 2-(46, 6, 1) desgns, weght 119; t s unknown f a desgn wth the latter parameters exsts. 17/21 P
The desarguesan plane P G 2,1 (F q ) does not contan subplanes of orders other than those from subfelds of F q, so the confguratons for n = 9 or 25 (1. and 2. of prevous page) cannot exst for the desarguesan case. It s conjectured that any non-desarguesan plane contans a Fano plane (see Neumann [Neu55]). Not all the known planes of order 25 have been checked for subplanes of order 4, but some are known not to have any; Clark [Cla00] has a survey of the known results. All known planes of square order have Baer subplanes. 18/21 P
References [AK92] [BSW03] E. F. Assmus, Jr and J. D. Key. Desgns and ther Codes. Cambrdge: Cambrdge Unversty Press, 1992. Cambrdge Tracts n Mathematcs, Vol. 103 (Second prntng wth correctons, 1993). Aart Blokhus, Tamás Szőny, and Zsuzsa Wener. On sets wthout tangents n Galos planes of even order. Des. Codes Cryptogr., 29:91 98, 2003. [CHKW03] K. L. Clark, L.D. Hatfeld, J. D. Key, and H. N. Ward. Dual codes of projectve planes of order 25. Advances n Geometry, 3:140 152, 2003. [CK99] [CKdR02] [Cla00] K. L. Clark and J. D. Key. Geometrc codes over felds of odd prme power order. Congr. Numer., 137:177 186, 1999. K. L. Clark, J. D. Key, and M. J. de Resmn. Dual codes of translaton planes. European J. Combn., 23:529 538, 2002. K. L. Clark. Improved bounds for the mnmum weght of the dual codes of some classes of desgns. PhD thess, Clemson Unversty, 2000. 19/21 P
[CO92] Terry Czerwnsk and Davd Oakden. The translaton planes of order twenty-fve. J. Combn. Theory, Ser. A, 59:193 217, 1992. [GW03] [KdR98] [KdR01] [KdR03] A. Gács and Zs. Wener. On (q + t, t)-arcs of type (0, 2, t). Des. Codes Cryptogr., 29:131 139, 2003. J. D. Key and M. J. de Resmn. Small sets of even type and codewords. J. Geom., 61:83 104, 1998. J. D. Key and M. J. de Resmn. Ternary dual codes of the planes of order nne. J. Statst. Plann. Inference, 95:229 236, 2001. J. D. Key and M. J. de Resmn. An upper bound for the mnmum weght of dual codes of fgueroa planes. J. Geom., 77:102 107, 2003. [KM90] Gábor Korchmáros and Francesco Mazzocca. On (q + t)-arcs of type (0, 2, t) n a desarguesan plane of order q. Math. Proc. Cambrdge Phlos. Soc., 108:445 459, 1990. [KN] J. D. Key and F. Ngwane. The mnmum weght of the dual 7-ary code of a projectve plane of order 49. In preparaton. 20/21 P
[Lm05] J. Lmbupasrporn. Partal permutaton decodng for codes from desgns and fnte geometres. PhD thess, Clemson Unversty, 2005. [MR95] Rudolf Mathon and Gordon F. Royle. The translaton planes of order 49. Des. Codes Cryptogr., 5:57 72, 1995. [Neu55] [Sac79] H. Neumann. On some fnte non-desarguesan planes. Arch. Math., VI:36 40, 1955. H. Sachar. The F p span of the ncdence matrx of a fnte projectve plane. Geom. Dedcata, 8:407 415, 1979. 21/21 P