Studying the locator polynomials of minimum

Studying the locator polynomials of minimum weight codewords of BCH codes. D. Augot 

P. Charpin y

N. Sendrier y

Abstract

We consider only primitive binary cyclic codes of length n = 2m ? 1. A BCH-code with designed distance  is denoted B (n;  ). A BCH-code is always a narrow-sense BCH-code. A codeword is identi ed with its locator polynomial, whose coecients are the symmetric functions of the locators. The de nition of the code by its zeros-set involves some properties for the power sums of the locators. Moreover the symmetric functions and the power sums of the locators are related with the Newton's identities. We rst present an algebraic point of view in order to prove or in rm the existence of words of a given weight in a code. The main tool is a symbolic computation software in exploring the Newton's identities. Our principal result is the true minimum distance of some BCH-codes of length 255 and 511, which were not known. In a second part, we study the codes B (n; 2h ? 1), h 2 [3; m ? 2]. We prove that the set of the minimum weight codewords of the BCH-code B (n; 2m?2 ? 1) equals the set of the minimum weight codewords of the punctured Reed-Muller code of length n and order 2, for any m. We give some Corollaries of this result.

 y

Universite Paris 6, UFR d'Informatique, LITP, 2 pl. Jussieu, 75251 Paris CEDEX 05, FRANCE INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex, FRANCE

1

1 Introduction In this paper, we deal with primitive binary cyclic codes. We are going to introduce a method for nding the true minimum distance of these codes. We will rst recall usual de nitions in section 2, as they are introduced in [10]. Our aim is to have a algebraic approach of the codewords in a cyclic code, which are studied through their locator polynomial. We describe the Newton's identities which allow us to study the properties of the locator polynomial of a codeword. In section 3, we will show how to use the Newton's identities. In fact we explore the identities in an progressive manner, using a symbolic computation software. We have two strategic options : trying to establish a contradiction to the existence of solutions to the identities ; or trying to nd an eective solution for the identities. This method enables us to complete the table of the minimum distance of the BCH codes in length 255, and to progress in the table of BCH codes in length 511. However the proofs are long and are given in the appendices A, B and C. In section 4 we give a description of the set of the minimum weight codewords of the BCH codes of length 2m ? 1 and designed distance 2m?2 ? 1 (Theorem 6). We prove that the locator polynomials of such codewords are, in fact, linearized polynomials. We obtain this result by studying the Newton's Identities associated to the minimum weight codewords of the BCH-codes of designed distance 2h ? 1, h 2 [2; m ? 1]. Some properties yield a complete characterization when h = m ? 2. When h 6= m ? 2, our proof involves an algorithm constructing cyclic codes whose minimumweight codewords have linearized locator polynomials.

2 Presentation and notations In this whole chapter we recall the usual conventions and notations used in [10].

2.1 The BCH codes and their minimum distance

We denote by GF (q) the Galois Field of order q, where q = 2m and by  a primitive n-root of unity in GF (q). Any cyclic code C of length n can be de ned by its generator polynomial whose roots are called the zeros of the code C . Thus we say that the de ning set of C is the set :

I (C ) = fi 2 [0::n ? 1] j i is a zero of C g We denote by cl(s) the cyclotomic class of s modulo n :

(1)

cl(s) = fs; 2s; 22s : : : ; 2m?1 s modulo 2m ? 1g (2) If i is a zero of C then 2i is also a zero of C , so we can see that I (C ) is a reunion of cyclotomic classes cl(s). Thus we can de ne the primitive narrow-sense BCH of length n of designed distance , denoted by B (n; ), as the cyclic code of length n whose de ning set is the union of the 2

cyclotomic classes cl(1); cl(2) : : : cl( ? 1). This terminology of \designed distance" is used because of the well known BCH-bound theorem : Theorem 1 If the de ning set of the cyclic code C contains a set of  ?1 consecutive integers (0 is treated consecutive to n-1), then the minimum distance of C is at least . So the code B (n; ) has minimum distance at least . But one will not be content with such a result. In general the designed distance is equal to the minimum distance, but we have no way to know systematically the true minimum distance. Of course there exists many other bounds for cyclic codes (J.H. van Lint deeply treats the subject in [12]), but still these are bounds and it is a dicult problem to nd the true minimum distance of a given BCH code, as soon as the length increases. The problem encountered in nding the true minimum distance is to work with the real structure of the nite eld GF (q), which deeply in uences the properties of cyclic codes, while bounds obtained with the properties of the de ning set of cyclic codes do not re ect the underlying algebraic structure of GF (q).

2.2 Mattson-Solomon polynomial and locator polynomial

De nition 1 The Mattson-Solomon polynomial of the word x = (x0; x1 : : : xn?1) is the polynomial of GF (q) :

A(z) = where

iX =n i=1

Ai = x(i) =

Aizn?i

j =X n?1 j =0

xj ij

(3) (4)

Remark :  A2i mod n = A2i  Ai+n = Ai So there is only one signi cant Ai for every cyclotomic class.

De nition 2 The locator polynomial (Z ) of a word x is the following polynomial : (Z ) =

iY =w i=1

(1 ? XiZ )

(5)

where the Xi are the elements of GF (q) which are not zeros of the Mattson-Solomon polynomial of x. They are called the locators of x.

3

De nition 3 The elementary symmetric functions of the locators X1; X2 : : : Xw are the i : 0 < i  w i = (?1)i P1k i0 then go to 8 ; 3. If i0 + r 2 T then put r := 0 and go to 2 ; 18

4. If r < 2h?1 , examine the identity I2i0+3r : if I2i0+3r : A2i0+r r = 0 then go to 7 else put T := T [ cl(i0 + r) and go to 7 ; 5. If r 2 Jh then go to 2 ; 6. If r > 2h?1 , examine the identity I2(i0+r): if I2(i0+r) : A2i0+r = 0 then go to 7 else put T := T [ cl(i0 + r) ; 7. Put r := 0 and A2i0j+r := 0 , for j 2 [0; m ? 1]; go to 2 ; 8. End. Example 2: m = 7; h = 4; thus i0 = 15 and J4 = f0; 8; 12; 14g. The code B (h) is the BCH-code of length 127 and designed distance 15. In accordance with Corollary 3, we have T4 = cl(19) [ cl(21). Using the algorithm, we obtain that the code C with de ning-set T = B (4) [ cl(19), satis es (RM4). Example 3: m = 8. 1) h = 4. The code B (4) is the BCH-code of length 255 and designed distance 15. We have:

T4 = cl(17) [ cl(19) [ cl(21) [ cl(25) : The algorithm produces: T = I (B (4)) [ cl(17) [ cl(19). 2) h = 5; J5 = f0; 16; 24; 28; 30; 31g. The code B (5) is the BCH-code of length 255 and designed distance 31. We have

T5 = cl(37) [ cl(39) [ cl(43) [ cl(45) [ cl(51) [ cl(53) : The algorithm produces: T = I (B (5)) [ cl(37) [ cl(39).

Acknowledgment The auhors wish to thank E.F. Assmus, G.D. Cohen and H.F. Mattson for enriching discussions and valuable suggestions.

19

Annex A

B (255; 61)

has minimum distance

>

61

We consider the Newton's identities Ir for 0 < r  n = 255, for the code B (255; 61), and for the weight  = 61. We want to prove that there exists no codeword of such weight. The non-null power sum symmetric functions of the code are : A61; A63; A85; A87; A91; A95; A111; A119; A127: And since 255 and 61 are relatively prime we can suppose A61 = 1 (the shift corresponds to a multiplication of each Ai by i). In the the case of a narrow-sense primitive BCH code, and for a weight equal to the designed distance, the Newton's identities Ir (9) for odd r from  + 2 to 2 ? 1 form a linear triangular system giving the i's for even i as polynomials depending on the non-null Ai's. Here the system consists of the 30 following equations : I63 : A63 + 2 = 0 I65 : A632 + 4 = 0 I67 : A634 + 6 = 0 I69 : A636 + 8 = 0 I71 : A638 + 10 = 0 I73 : A6310 + 12 = 0 I75 : A6312 + 14 = 0 I77 : A6314 + 16 = 0 I79 : 1 + A6316 + 18 = 0 I81 : 2 + A6318 + 20 = 0 I83 : 4 + A6320 + 22 = 0 I85 : A85 + 6 + A6322 + 24 = 0 I87 : A87 + A852 + 8 + A6324 + 26 = 0 I89 : A872 + A854 + 10 + A6326 + 28 = 0 I91 : A91 + A874 + A856 + 12 + A6328 + 30 = 0 I93 : A487 + A912 + A876 + A858 + 14 + A6330 + 32 = 0 I95 : A95 + A4872 + A914 + A878 + A8510 + 16 + A6332 + 34 = 0 I97 : A952 + A4874 + A916 + A8710 + A8512 + 18 + A6334 + 36 = 0 I99 : A954 + A4876 + A918 + A8712 + A8514 + 20 + A6336 + 38 = 0 I101 : A956 + A4878 + A9110 + A8714 + A8516 + 22 + A6338 + 40 = 0 I103 : A958 + A48710 + A9112 + A8716 + A8518 + 24 + A6340 + 42 = 0 I105 : A9510 + A48712 + A9114 + A8718 + A8520 + 26 + A6342 + 44 = 0 4 I107 : A32 91 + A9512 + A8714 + A9116 + A8720 + A8522 + 28 + A6344 + 46 = 0 4 I109 : A491 + A32 912 + A9514 + A8716 + A9118 + A8722 + A8524 + 30 + A6346 +48 = 0 20

4 I111 : A111 + A4912 + A32 914 + A9516 + A8718 + A9120 + A8724 + A8526 + 32 +A6348 + 50 = 0 4 I113 : A1112 + A4914 + A32 916 + A9518 + A8720 + A9122 + A8726 + A8528 + 34 +A6350 + 52 = 0 4 I115 : A1114 + A4916 + A32 918 + A9520 + A8722 + A9124 + A8728 + A8530 + 36 +A6352 + 54 = 0 16 4 I117 : A87 + A1116 + A4918 + A32 9110 + A9522 + A8724 + A9126 + A8730 + A8532 +38 + A6354 + 56 = 0 4 32 4 I119 : A119 + A16 872 + A1118 + A9110 + A9112 + A9524 + A8726 + A9128 + A8732 +A8534 + 40 + A6356 + 58 = 0 4 4 32 I121 : A1192 + A16 874 + A11110 + A9112 + A9114 + A9526 + A8728 + A9130 + A8734 +A8536 + 42 + A6358 + 60 = 0

which gives us the following values for the i's :

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

:= := := := := := := := := := := := := := := := := := := := := := :=

A63 A263 A363 A463 A563 A663 A763 A863 1 + A963 A10 63 A263 + A11 63 12 A85 + A63 A87 + A463 + A13 63 A85A263 + A14 63 2 A91 + A87A63 + A663 + A15 63 16 4 4 A87 + A85A63 + A63 A95 + A91A263 + A87A463 + A863 + A17 63 4 2 6 18 A87A63 + A85A63 + 1 + A63 19 A95A263 + A91A463 + A87A663 + A10 63 + A63 + A63 A487A463 + A85A863 + A20 63 4 6 21 A95A63 + A91A63 + A87A863 + A12 63 + A63 4 22 A487A663 + A85A10 63 + A63 + A63 6 8 10 14 5 23 A32 91 + A95A63 + A91A63 + A87A63 + A63 + A63 + A63 21

48 50 52 54 56 58 60

:= := := := := := :=

24 A491 + A487A863 + A285 + A85A12 63 + A63 2 8 10 12 16 2 25 A111 + A32 91A63 + A95A63 + A91A63 + A87A63 + A63 + A63A85 + A63 4 2 4 10 14 A863 + A287 + A26 63 + A91A63 + A87A63 + A85A63 4 27 10 12 14 2 2 18 1 + A963 + A32 91A63 + A63 + A95A63 + A91A63 + A87A63 + A63A87 + A111A63 + A63 28 4 12 2 4 16 A491A463 + A16 87 + A63 + A87A63 + A85A63 + A85A63 32 6 12 14 16 2 5 20 A111A463 + A119 + A29 63 + A91A63 + A95A63 + A91A63 + A87A63 + A85A63 + A63 16 2 4 6 4 14 30 12 A85 + A291 + A287A463 + A85A18 63 + A87A63 + A91A63 + A87A63 + A63 + A63

The other values are 0 = 1, by de nition, and i = 0 for odd i, given by the  rst identities. After substitution of the i's by their values, the remaining equations are sorted in increasing size (number of monomials) order : 186, 190, 188, 194, 198, 192, 202, 123, 184, 189, 191, 196, 206, 254, 127, 195, 200, 210, 135, 193, 187, 199, 214, 125, 131, 204, 252, 197, 222, 203, 238, 129, 139, 208, 250, 143, 201, 218, 133, 137, 226, 230, 246, 248, 234, 185, 212, 242, 207, 141, 181, 216, 236, 244, 151, 183, 205, 220, 232, 147, 224, 159, 179, 211, 240, 149, 175, 155, 145, 157, 209, 173, 163, 167, 228, 153, 171, 215, 253, 251, 165, 169, 161, 177, 213, 223, 239, 247, 249, 243, 217, 235, 237, 245, 219, 221, 229, 231, 233, 227, 225, 241 We will proceed as follow :  we successively check the equations in the order given above, up to a \solvable" one.  After solving one equation, we restart from the beginning. (at each stage we substitute all the known Ai's in the current equation, and we simplify it as much as possible) We give here in the resolution order all the \solvable" equations, and the way we used them. 16 4 4 8 4 16 32 4 I186 : A887 + A285A863 + A85A20 63 + A87A63 + A91A63 + A87A63 + A63 + A95 = 0

) A95 := A287 + A285A263 + A85A563 + A487A63 + A91A263 + A87A463 + A863 I188 : A887A63 + A291A363 + A287A763 + A285 + A285A963 + A91 + A87A263 + A663 + A15 63 +A127 = 0

) A127 := A887A63 + A291A363 + A287A763 + A285 + A285A963 + A91 + A87A263 + A663 + A1563 I194 : A385 + 1 = 0 ) A85 6= 0 22

I198 : A85A287 + A263 = 0 ) A87 := A63A85 2 6 4 21 2 142 130 128 2 I187 : 1 + A91A85A18 63 + A85A91A63 + A91A63 + A63 + A63 + A63 A85 + A111A63 3 2 39 8 6 9 2 33 4 3 2 2 +A16 91A63 + A85A63 + A119 + A91A63 + A91A63 + A91A63 + A85A91 36 15 30 3 24 2 15 4 +A285A21 63 + A63 + A63A85 + A91A63 + A91 + A85A63 + A85A63A91 9 4 4 12 2 6 5 6 2 4 6 +A291A21 63A85 + A63A85A91 + A91A63 + A91A63 + A91A63 + A85A91A63 45 12 3 2 +A285A30 63 + A63 + A91A85 + A91A63 + A63A85 = 0 160 192 32 2 66 64 225 128 ) A119 := 1 + A16111A6463 + A85A228 63 + A91 A63 + A91A85A63 + A63 + A85A63 A91

195 2 96 64 132 32 195 2 225 2 +A80 63A85 + A63 A85 + A91A63 + A85A63 + A91A63 + A91A85 + A63 96 192 33 64 162 2 96 33 2 128 128 192 +A128 91 A63 + A91 A63 + A91A63 A85 + A63A85 + A63A85A91 + A85A91 A63 128 129 165 32 192 3 2 209 128 162 129 +A32 91A63 + A63 + A91 A63 + A63 + A85A91A63 + A63A85 + A91 A63 36 162 64 192 96 32 2 +A64 91A63 + A85A63 + A91A63 + A91 + A91A85

4 10 31 3 4 4 25 2 22 I189 : A291A34 63 + A91A85A63 + A91A63 + A63A91 + A91A63 + A85A63 + A85A63 128 2 2 31 5 7 4 13 7 2 +A91A85A19 63 + A63A91 + A91A63 + A91A63 + A85A63A63 + A63 + A63 A91 2 4 2 2 128 2 4 22 13 4 +A37 63 + A111 + A91A63 + A91A63 + A63 A85 + A91A63A85 + A85A63 3 134 2 40 6 10 46 2 7 +A85A16 63 + A63 + A63 + A85A63 + A91A63 + A63 + A85A91A63 2 4 7 2 22 4 +A285A16 63A91 + A91A63A85 + A85A91A63 + A63A91A85 = 0 2 130 64 196 128 64 2 ) A111 := A91A285A463 + A133 63 A85 + A91A85A63 + A91A85A63 + A91 A63A85

129 130 2 10 139 128 64 70 193 +A285A64 91A63 + A63A85 + A63 + A63 + A63 + A85A63 + A91 A63 73 64 64 32 2 2 193 2 133 +A128 91 A63 A85 + A85A91A63 + A63A85 + A63A85 + A63A91A85 + A63 65 193 2 199 64 67 161 +A91A63 + A85A463 + A133 63 A91 + A91A63 + A85A63 + A91A63 + A63 64 192 192 32 64 193 128 128 136 67 199 +A64 91A63 + A91A63 + A91 A63 + A63 A91 + A63A91 + A63 + A63A91

I199 : A16 91 = 0 ) A91 := 0 I203 : 1 = 0

Annex B

B (255; 59)

has minimum distance

2 >

59

We consider the Newton's identities Ir for 0 < r  n = 255, for the code B (255; 59), and for the weight  = 59. We want to prove that there exists no codeword of such weight. 23

The non-null power sum symmetric functions of the code are : A59; A61; A63; A85; A87; A91; A95; A111; A119; A127: And since 255 and 59 are relatively prime we can suppose A59 = 1. We will rst solve the linear triangular system giving the i's for even i as polynomials depending on the non-null Ai's. The system consists of the 29 following equations : I61 : A61 + 2 = 0 I63 : A63 + A612 + 4 = 0 I65 : A632 + A614 + 6 = 0 I67 : A634 + A616 + 8 = 0 I69 : A636 + A618 + 10 = 0 I71 : A638 + A6110 + 12 = 0 I73 : A6310 + A6112 + 14 = 0 I75 : A6312 + A6114 + 16 = 0 I77 : A6314 + A6116 + 18 = 0 I79 : A64 61 + A6316 + A6118 + 20 = 0 I81 : A64 612 + A6318 + A6120 + 22 = 0 I83 : A64 614 + A6320 + A6122 + 24 = 0 I85 : A85 + A64 616 + A6322 + A6124 + 26 = 0 I87 : A87 + A852 + A64 618 + A6324 + A6126 + 28 = 0 I89 : A872 + A854 + A64 6110 + A6326 + A6128 + 30 = 0 I91 : A91 + A874 + A856 + A64 6112 + A6328 + A6130 + 32 = 0 4 I93 : A87 + A912 + A876 + A858 + A64 6114 + A6330 + A6132 + 34 = 0 4 I95 : A95 + A872 + A914 + A878 + A8510 + A64 6116 + A6332 + A6134 + 36 = 0 4 I97 : A952 + A874 + A916 + A8710 + A8512 + A64 6118 + A6334 + A6136 + 38 = 0 4 I99 : A954 + A876 + A918 + A8712 + A8514 + A64 6120 + A6336 + A6138 + 40 = 0 4 I101 : A956 + A878 + A9110 + A8714 + A8516 + A64 6122 + A6338 + A6140 + 42 = 0 4 I103 : 1 + A958 + A8710 + A9112 + A8716 + A8518 + A64 6124 + A6340 + A6142 +44 = 0 I105 : 2 + A9510 + A48712 + A9114 + A8718 + A8520 + A64 6126 + A6342 + A6144 +46 = 0 32 I107 : A91 + 4 + A9512 + A48714 + A9116 + A8720 + A8522 + A64 6128 + A6344 +A6146 + 48 = 0 4 4 64 I109 : A91 + A32 912 + 6 + A9514 + A8716 + A9118 + A8722 + A8524 + A6130 + A6346 +A6148 + 50 = 0 4 64 I111 : A111 + A4912 + A32 914 + 8 + A9516 + A8718 + A9120 + A8724 + A8526 + A6132 +A6348 + A6150 + 52 = 0 24

4 I113 : A1112 + A4914 + A32 916 + 10 + A9518 + A8720 + A9122 + A8726 + A8528 +A64 6134 + A6350 + A6152 + 54 = 0 4 I115 : A1114 + A4916 + A32 918 + 12 + A9520 + A8722 + A9124 + A8728 + A8530 +A64 6136 + A6352 + A6154 + 56 = 0 16 4 I117 : A87 + A1116 + A4918 + A32 9110 + 14 + A9522 + A8724 + A9126 + A8730 +A8532 + A64 6138 + A6354 + A6156 + 58 = 0 16 4 I119 : A119 + A872 + A1118 + A49110 + A32 9112 + 16 + A9524 + A8726 + A9128 +A8732 + A8534 + A64 6140 + A6356 + A6158 = 0

which gives us the following values for the i's :

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

:= := := := := := := := := := := := := := := := := :=

38 := 40 := 42 := 44 :=

A61 A261 + A63 A361 A63A261 + A263 + A461 A61A263 + A561 A363 + A63A461 + A661 A761 A463 + A263A461 + A63A661 + A861 A61A463 + A263A561 + A961 5 3 4 8 2 4 10 A64 61 + A63 + A63A61 + A63A61 + A61A63 + A61 A361A463 + A11 61 6 + A2 A8 + A2 A5 + A A10 + A12 + A A66 63 61 61 61 63 63 61 63 61 6 2 9 13 A85 + A61A63 + A63A61 + A61 14 12 3 8 7 68 2 A87 + A64 61A63 + A61 + A63 + A63A61 + A63A61 + A61 A85A261 + A15 61 16 14 2 12 4 8 8 A91 + A87A261 + A70 61 + A63 + A63A61 + A63A61 + A63A61 + A61 17 A487 + A85A263 + A85A461 + A61A863 + A463A961 + A263A13 61 + A61 5 8 68 2 64 4 4 2 2 9 72 A95 + A18 61 + A61 + A63 + A91A61 + A87A63 + A87A61 + A61A63 + A61A63 + A63A61 16 2 8 4 10 +A363A12 61 + A63A61 + A61A63 + A63A61 19 4 2 A85A661 + A361A863 + A463A11 61 + A61 + A87A61 6 8 2 16 2 9 5 10 4 8 66 4 2 A128 61 + A63A61 + A63A61 + A61A63 + A63A61 + A61A63 + A61A63 + A95A61 10 20 6 18 +A91A263 + A91A461 + A74 61 + A63 + A61 + A87A61 + A63A61 5 8 A85A263A461 + A487A263 + A487A461 + A85A463 + A85A861 + A663A961 + A263A17 61 + A61A63 129 21 +A61A10 63 + A61 + A61 22 76 11 2 4 6 1 + A87A263A461 + A63A128 61 + A61 + A61 + A63 + A95A63 + A95A61 + A91A61 6 72 2 7 8 3 16 4 9 20 +A87A463 + A87A861 + A64 61A63 + A61A63 + A63A61 + A63A61 + A61A63 + A63A61 +A661A863 25

4 6 10 46 := A85A261A463 + A761A863 + A23 61 + A87A61 + A85A61 24 132 78 2 32 6 4 48 := A91A263A461 + A87A261A463 + A12 63 + A61 + A61 + A61 + A61 + A91 + A95A61 + A91A63 22 4 16 4 10 2 20 6 9 +A91A861 + A87A10 61 + A63A61 + A63A61 + A61A63 + A63A61 + A61A63 12 4 17 5 10 2 21 2 8 4 2 4 50 := A491 + A85A12 61 + A61A63 + A63A61 + A61A63 + A63A61 + A85A63A61 + A87A63A61 133 4 4 4 8 6 +A25 61 + A61 + A87A63 + A87A61 + A85A63 8 72 4 76 2 24 132 5 16 4 11 52 := A461 + A263 + A64 61A63 + A61A63 + A61A63 + A63A61 + A63A61 + A63A61 + A61A63 2 12 4 18 80 13 26 2 4 2 4 +A363A20 61 + A61A63 + A63A61 + A61 + A63 + A61 + A95A63A61 + A91A61A63 32 2 8 6 10 2 +A87A263A861 + A95A463 + A87A12 61 + A91A61 + A95A61 + A87A63 + A91A61 + A85 +A111 2 3 12 4 19 27 4 10 4 2 54 := A487A261A463 + A85A14 61 + A61A85 + A61A63 + A63A61 + A61 + A87A61 + A91A61 32 2 136 28 2 2 82 56 := A91A263A861 + A95A261A463 + A14 63 + A87 + A61 + A61 + A61 + A111A61 + A91A63 4 10 6 12 14 128 4 66 8 74 4 +A32 91A61 + A95A61 + A91A63 + A91A61 + A87A61 + A61 A63 + A61A63 + A61A63 6 16 2 13 5 18 26 6 +A63A285 + A263A24 61 + A63A61 + A61A63 + A63A61 + A63A61 + A61 2 16 129 4 14 4 2 8 4 8 58 := A85A263A12 61 + A85A63A61 + A87A63A61 + A87 + A61 A63 + A61A63 + A61A87 6 17 137 29 4 2 4 4 4 6 4 12 +A263A25 61 + A63A61 + A61 + A61 + A91A63 + A91A61 + A87A63 + A87A61 +A85A863 + A85A16 61

The other values are 0 = 1, by de nition, and i = 0 for odd i, given by the rst identities. After substitution of the i's by their values, the remaining equations will be sorted in increasing size (number of monomials) order : 180, 188, 196, 192, 186, 184, 204, 200, 190, 252, 189, 119, 178, 182, 187, 212, 121, 194, 208, 220, 236, 198, 125, 185, 244, 191, 193, 202, 248, 197, 216, 228, 224, 123, 133, 195, 129, 206, 232, 250, 240, 205, 137, 141, 201, 181, 183, 246, 254, 218, 127, 199, 210, 173, 179, 203, 157, 253, 131, 214, 177, 149, 171, 234, 249, 139, 145, 153, 222, 242, 230, 135, 251, 169, 238, 245, 155, 165, 213, 221, 209, 207, 237, 241, 217, 226, 147, 175, 161, 143, 243, 163, 151, 211, 167, 247, 233, 235, 219, 159, 229, 225, 239, 227, 215, 231, 223 We will proceed as follow :  we successively check the equations in the order given above, up to a \solvable" one.  After solving one equation, we restart from the beginning. (at each stage we substitute all the known Ai's, and we show the most simple equation possible) We will rst show that A61 6= 0 Suppose that A61 = 0, then :

I196 : A385 = 0 ) A85 := 0; 26

I208 : A687 = 0 ) A87 := 0; I236 : 1 = 0;

so A61 6= 0. We give here in the resolution order all the \solvable" equations, and the way we used them. 3 2 16 12 4 16 I180 : A861A85A463 + A85A863 + A461A491 + A29 61 + A61A85 + A61A85 + A61A87 + A87 = 0 2 2 70 191 2 3 254 4 ) A91 := A61A85A63 + A254 61 A85A63 + A61A87 + A61 + A61 A85 + A61A85 + A61 A87 8 139 4 I196 : A295A361 + A887A61A263 + A285A61A663 + A287A361A463 + A385 + A131 61 A63 + A61 A63 4 5 2 4 +A130 61 A87 + A61A85A63 = 0 3 254 4 254 3 2 2 64 4 ) A95 := A126 61 A85 + A61 A87A63 + A61 A85A63 + A87A63 + A61A85A63 + A61A63 191 2 2 +A68 61A63 + A61 A87

I192 : A385 = 0 ) A85 = 0 I200 : 1 = 0

Annex C

B (511; 123)

has minimum distance

2 >

123

We consider the Newton's identities for 0 < i  n = 511 for the code B (511; 123), and for the weight  = 123. We want to prove that there exists no codeword of such weight. The non-null power sum symmetric functions of the code are :

A123; A125; A127; A171; A175; A183; A187; A191; A219; A223; A239; A255: And since 511 and 123 are relatively prime we can suppose A123 = 1. We will show for this code a shorter proof. The complete proof would be too long to appear here. We will rst solve the linear triangular system giving the i's for even i as polynomials depending on the non-null Ai's. The i's for odd i are null. We consider that the i's have been substituted in the equations. Furthermore we will suppose A125 6= 0 (when A125 = 0, we found a contradiction). 27

We give here the equations we used for the resolution, and the way we used them. 8 13 32 5 4 8 22 4 6 16 4 6 I372 : A63 125 + A171A125 + A171A125 + A171A127A125 + A171A125A127 + A187 A125 2 4 14 4 38 16 8 18 16 2 16 +A16 175A125 + A183A125 + A171A125 + A171A127A125 + A171A125A127 4 26 16 34 4 30 64 2 8 64 4 10 +A16 171A127A125 + A171A125 + A175A125 + A171A125A127 + A171A127A125 18 +A64 171A125 = 0 385 2 383 8 2 4 4 3 4 2 ) A187 := A142 125 + A125A171 + A125A171 + A171A127A125 + A171A127 + A125A171A127 4 2 8 6 510 4 4 +A510 125A175 + A183A125 + A171A125 + A175A125 + A125A171A127 16 2 16 16 3 +A5125A4171A127 + A4171A7125 + A510 125A171A127 + A125A171A127 + A171A125 4 257 4 282 4 258 16 4 257 274 8 I388 : A8171A10 127 + A125A127 + A175A125 + A125A127 + A125A127 + A171A125A127 8 6 8 32 6 2 2 8 8 2 +A2175A4127A10 125 + A171A127A125 + A171A127 + A175A125A127 + A175A127 4 12 8 4 +A2183A2125A4127 + A2191 A2125 + A4125A32 171A127 + A125A171A127 2 4 +A14 125A171A127 = 0 4 5 128 8 383 2 136 4 140 2 383 2 2 ) A191 := A510 125A171A127 + A125A127 + A125A175 + A125A127 + A125A127 + A125A171A127 510 4 4 16 3 +A175A2127A4125 + A3125A4171A3127 + A510 125A171A127 + A175A127 + A125A175A127 2 4 2 5 6 2 +A183A2127 + A16 171A125A127 + A171A127A125 + A125A171A127

I392 : A2125 + A6171 = 0 ) A171 := A341 125 341 2 I404 : A2127 + A340 125A175 = 0 ) A175 := A127A125

I412 : A97 125 + 1 = 0 ) A125 := 1 I420 : A2127 + A2183 + A4183 = 0 ) A127 := A183 + A2183 I428 : 1 = 0

2

References [1] E.F. Assmus and J.D. Key. Ane and projective planes. Discrete Mathematics, 83:161{ 187, 1990. [2] P. Charpin. Codes cycliques etendus anes-invariants et anticha^ines d'un ensemble partiellement ordonne. Discrete Mathematics, 80:229{247, 1990. 28

[3] G. Cohen. On the minimum distance of some BCH codes. IEEE Transaction on Information Theory, 26, 1980. [4] J-L. Dornstetter. Quelques resultats sur les codes BCH binaires en longueur 255. ENSTA stage report, Annex, July 1982. [5] H.J. Helgert and R.D. Stina. Shortened BCH codes. IEEE Transaction on Information Theory, pages 818{820, November 1973. [6] T. Kasami and S. Lin. Some results on the minimum weight of primitive BCH codes. IEEE Transaction on Information Theory, pages 824{825, November 1972. [7] T. Kasami, S. Lin, and W.W. Peterson. Some results on cyclic codes which are invariant under the ane group and their applications. Information and Control, 11:475{496, 1967. [8] T. Kasami, S. Lin, and W.W. Peterson. New generalisations of the Reed-Muller codes { Part I: Primitive codes. IEEE Transaction on Information Theory, 14(2):189{199, March 1968. [9] T. Kasami and N. Tokura. Some remarks on BCH bounds and minimum weights of binary primitive BCH codes. IEEE Transaction on Information Theory, 15(3):408{413, May 1969. [10] F.J. MacWilliams and N.J.A. Sloane. The Theory of Error Correcting Codes. NorthHolland, 1986. [11] W. W. Peterson. Error-Correcting Codes. MIT Press, 1961. [12] J.H. van Lint and R.M. Wilson. On the minimum distance of cyclic codes. IEEE Transaction on Information Theory, 32(1):23, January 1986.

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