Sixteen New Linear Codes With Plotkin Sum

SUBMITTED TO IEEE TRANS. I NFORMA TION THEOR Y 1 Sixteen Ne w Linear Codes W ith Plotkin Sum Fernando Hernando and Diego Ruano Abstract —Sixteen new linear codes are pres ented: three of them impro ve the lower bounds on the minimu m distance fo r a linear c ode and the re st are an explicit co nstruction of unk nown codes attaining the lower bounds on the minimu m distance. They are constructed using th e Plotkin sum of two linear codes, also called ( u | u + v ) construction. Th e computations have b een achiev ed using an exhaustiv search. Index T erms —Linear Code, Plotkin Sum, ( u | u + v ) Construc- tion, Minimu m Dist ance’ s Lower Bound. I . I N T RO D U C T I O N P LO TKIN sum, also called ( u | u + v ) constructio n and bar produ ct, is a classic tool to construct co des fro m codes already known. It was intr oduced in 1 960 by M. Plotkin [1] and then rediscovered in [2]. Howe ver , we show tha t this construction can still be used to obtain codes that imp rove the linear c odes’ bounds. W e ha ve con sidered the tables of linear co des listed in [3]. W e obtain three codes over F 4 , that im prove the lower bo unds of the m inimum distance and thirteen codes, over F 3 and F 4 , who se existence was kn own but who se con struction was unknown. W e also show that Plotkin bound can be u sed to obtain a sign ificant amount of codes listed in [3], sometimes in a simp lier way . I I . P L O T K I N S U M Let C 1 , C 2 be two linear codes in F n q with parameters [ n, k 1 , d 1 ] and [ n, k 2 , d 2 ] respectively . Th e Plotkin sum of C 1 and C 2 is C = { ( u, u + v ) | u ∈ C 1 , v ∈ C 2 } ⊂ F 2 n q One can see in [4] that C is a linear code with parameter s [2 n, k 1 + k 2 , min { 2 d 1 , d 2 } ] . I I I . N E W C O D E S In [3] one can find a list with the bou nds for the minimum distance of linear co des over F q , with q = 2 , 3 , 4 , 5 , 7 , 8 , 9 , with length and dimension lower than or equal to 256 , 243 , 256 , 13 0 , 100 , 130 , 130 , respectively . W e hav e conside red th e minimum distance of the Plotkin sums of two c odes whose length n is in th e first half o f the table (we cannot compare the sum of codes in the second half), for instance for q = 4 , n = 1 , . . . , 12 1 . W e have com pared The work of F . Hernando is supporte d in part by the Claude Shannon Institut e, Science Foundatio n Irelan d Grant 06/MI/006 (Ireland) and MEC MTM2007-64704 an d by Junta de CyL V A02 5A07 (Spain). T he w ork o f D. Ruano is supported in pa rt by DT U, H.C. O ersted post do c. grant ( Denmark) and MEC MTM2007-64704 and by Junta de CyL V A065A07 (Spai n) F . Hernando is with the Department of Mathematics, Univ ersity Colle ge Cork, Irel and, e-mail : F .Hernando@uc c.ie D. Ruano is with the Department of Mathematics, T ec hnical Uni v ersity of Denmark, Matematiktor vet , Building 303, DK-2800, Ly ngby , Denmark, e-mail: D.Ru ano@mat.dtu .dk their minimu m distan ce with the cod es of length 2 n in [ 3]. This computation can be easily and fas t achieved w ith a simple computer pr ogram. W e o btained the f ollowing new cod es: Three cod es over F 4 which im prove the lower bound s: • The plo tkin sum o f the co des with paramete rs [6 3,53, 6] and [63 ,42,1 2] in [3] gives a [126,95 ,12] code, the lower bound was 11. • The plo tkin sum o f the co des with paramete rs [6 4,54, 6] and [64 ,43,1 2] in [3] gives a [128,97 ,12] code, the lower bound was 1 1. Fu rthermo re, considering a s hortenin g [4], one obtains a [127,96, ≥ 12] code , th e lower bound was also 11. W e have obtaine d thr ee codes over F 3 that give an explicit construction , wh ich was u nknown, f or th e lower bou nd: • The plo tkin sum o f the co des with paramete rs [6 2,46, 8] and [62,32 ,16] in [ 3] gives a [1 24,78 ,16] code . And considerin g a shorten ing, one obtain s a [123,77 , ≥ 16] code. • The plo tkin sum o f the co des with paramete rs [6 3,47, 8] and [ 63,32 ,17] in [3] gives a [12 6,79,1 6] code. W e have ob tained ten codes over F 4 that give an exp licit construction , un known so far, fo r the lower bound : • The plo tkin sum o f the co des with paramete rs [5 2,38, 8] and [5 2,25,1 6] in [3] gi ves a [104,63 ,16] cod e. And, considerin g a shorten ing, one obtain s a [103,62 , ≥ 16] code. • The plo tkin sum o f the co des with paramete rs [5 3,39, 8] and [5 3,26,1 6] in [3] gi ves a [106,65 ,16] cod e. And, considerin g a shorten ing, one obtain s a [105,64 , ≥ 16] code. • The plo tkin sum o f the co des with paramete rs [5 4,40, 8] and [5 4,27,1 6] in [3] gi ves a [108,67 ,16] cod e. And, considerin g a shorten ing, one obtain s a [107,66 , ≥ 16] code. • The plo tkin sum o f the co des with paramete rs [6 1,51, 6] and [ 61,40 ,12] in [3] gives a [12 2,91,1 2] code. • The plo tkin sum o f the co des with paramete rs [6 2,52, 6] and [6 2,41,1 2] in [3] gi ves a [124,93 ,12] cod e. And, considerin g a shorten ing, one obtain s a [123,92 , ≥ 12] code. • Considering a shorten ing o f the [126 ,95,12 ] code above, one o btains a [125 ,94, ≥ 12] One can easily obtain the codes with the compu ter alge- bra sy stem Magma [ 5]. F or instance we constru ct the code [122,9 1,12] over F 4 , by considering the Plotkin sum of the correspo nding two code s listed in [3]: > F:=GF(4); > P:=Polynomia lRing(F); > a:=F.1; SUBMITTED TO IEEE TRANS. INFORMA TION T HEOR Y 2 > TMP1:=BCHC ode(F, 63, 5); > TMP2:=Exte ndCode(TMP1, 1); > C1:=Shorte nCode(TMP2,{ 62 .. 64 }); > TMP3:=Cycl icCode(65,xˆ 21+a * xˆ20+a * xˆ19+ a * xˆ18+aˆ2 * xˆ15+aˆ2 * xˆ14+aˆ2 * xˆ12+xˆ11+ xˆ10+aˆ2 * xˆ9+aˆ2 * xˆ7+aˆ2 * xˆ6+a * xˆ3+a * xˆ2+ a * x+1); > C2:=Shorte nCode(TMP3,{ 62 .. 65 }); > C:=Plotkin Sum(C1,C2); Finally , we remar k that a significant amount of codes in [3] can be obtained using t he Plotkin s um. W e comp are the bounds for the minimum distance of linear code s with even len gth listed in [3] and sho w how many of them can be obtained using this su m. Som etimes, this con struction is simpler th an the one in [3] (conside ring several sho rtenings, p uncturin gs, or p arity ch eck bits, ...). q # in [3], n even # Plotkin Sum % 2 16512 2676 16.20 3 14762 1681 11.38 4 16512 1350 8.17 5 4290 495 11.5 3 7 2550 354 13.8 8 8 4290 454 10.5 8 9 4290 431 10.0 4 I V . C O N C L U S I O N Nine new linear codes are presente d. T hey are obtained using the Plotkin sum, theref ore we suggest that whenever a new linear cod e of leng th n is obtaine d, o ne should ch eck if it is possible to obtain a code o f length 2 n , which improves the lower distance b ounds, b y using the Plotkin sum o f that code and the codes of length n in [3]. Further more, t his computation can b e easily achieved. A C K N OW L E D G M E N T The authors would like to thank M. Greferath for his course at Claude Shan non Institute and T . Høholdt for help ful comments on this paper . R E F E R E N C E S [1] M. Plotkin, “Binary codes with specified minimum distance, ” IRE T ran s. , vol. IT -6, pp. 445–450, 1960. [2] N. J. A. Sloane and D. S. White head, “Ne w fa mily of s ingle-e rror correct ing code s, ” IEEE T rans. Information Theory , vol. IT -16, pp. 717– 719, 1970. [3] M. Grassl, “Bounds on the minimum distance of linear codes, ” Online av ailable at http://www .codetabl es.de , 2007, accessed on 2008-04-21. [4] F . Macwill iams and N. Sloane, The Theory of Error -Corr ectin g Codes , ser . North-Holland mathematica l library . No rth-Holla nd, 1977, vol. 16. [5] W . Bosma, J. Cannon, and C. Playoust, “The magma algeb ra system. I. the user language, ” J . Symbolic Comput. , vol. 24(3-4), pp. 235–265, 1997.