Formally self-dual additive codes over F 4

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1 Formally self-dual additive codes over F Sunghyu Han School of Liberal Arts, Korea University of Technology and Education, Cheonan 0-708, South Korea Jon-Lark Kim Department of Mathematics, University of Louisville, Louisville, KY 09, USA Abstract Additive codes over F have been of great interest due to their application to quantum errorcorrection. As another application, we introduce a new class of formally self-dual additive codes over F, which is a natural analogue of binary formally self-dual codes and is missing in the study of additive codes over F. In fact, Gulliver and Östergård (00) considered formally self-dual linear codes over F of even lengths, and Choie and Solé (008) suggested to classify formally self-dual linear codes over F of odd lengths in order to study lattices from these codes. These motivate our study on formally self-dual additive codes over F. In this paper, we define extremal and near-extremal formally self-dual additive codes over F, classify all extremal codes, and construct many near-extremal codes. We discuss a general method (called the weak balance principle) to construct such codes. We conclude with some open problems. Key words: Additive codes; balance principle, extremal codes; formally self-dual additive codes; near-extremal codes 1. Introduction Binary self-dual codes and additive self-dual codes over F have common properties such as Type I, Type II, shadow codes, and s-extremal codes (Han and Kim, 008), (Rains and Sloane, 1998). Binary formally self-dual codes are defined as a class of binary codes whose weight enumerators are the same as those of their dual codes. Hence binary formally self-dual codes include the class of binary self-dual codes, and their weight enumerators are combinations of Gleason polynomials of Type I (Kennedy and Pless, 199). One of the motivations for studying binary formally self-dual codes is that some binary addresses: sunghyu@kut.ac.kr (Sunghyu Han), jl.kim@louisville.edu (Jon-Lark Kim). URL: jlkim (Jon-Lark Kim). Preprint submitted to Elsevier 1 July 009

2 formally self-dual codes (e.g., at lengths 10, 18 (Kennedy and Pless, 199)) have better minimum distances than any binary self-dual codes of the same length. This observation leads us to ask our main question of the paper. Is there a class of formally self-dual additive codes over F? If so, classify such codes and find their highest minimum distances using their extremal or near-extremal weight enumerators. We hope that this will open a new research area in the study of additive codes. Additive codes have been less studied by researchers due to the lack of the linear structure in general. Additive self-dual codes over F often have better minimum distances than any Hermitian self-dual linear codes over F. For example, there is no linear Hermitian self-dual [1, 6, 6] code over F (e.g., (Huffman, 00), (Rains and Sloane, 1998)). However there is an additive trace self-dual (1, 1, 6) code over F, called the dodecacode (Calderbank et. al., 1998). Additive codes over F can be generalized to additive codes over F p for any prime p. However since it will require more computational work, in this paper we focus on additive codes over F. In fact, there has been some attempt to do this problem. For example, Gulliver and Östergård (Gulliver and Östergård, 00) constructed formally self-dual linear codes over F of even lengths by classifying all optimal linear [n, n/, d] codes over F of even lengths up to 18. Clearly the existence of a linear code over F implies the existence of an associated additive code over F. On the other hand, Choie and Solé (Choie and Solé, 008, Section 6) suggested to classify formally self-dual linear codes over F of odd lengths in order to study lattices from these codes. These also motivate our current study on formally self-dual additive codes over F. Furthermore, it is worth mentioning that the class of formally self-dual additive codes can be put together with the four types of classical (formally) self-dual codes (Rains and Sloane, 1998) (i.e., Type I binary formally self-dual codes, Type II binary self-dual codes, Type III ternary self-dual codes, and Type IV Hermitian self-dual codes) since the weight enumerators of these five classes are generated by two Gleason polynomials. Our contribution in this paper is to introduce a new class of formally self-dual additive codes over F, classify extremal ones, and construct near-extremal ones by using the weak balance principle which simplifies the calculation. We also find an upper bound on the highest minimum distance of these codes. This paper is organized as follows. Section gives a brief introduction to additive codes over F and defines extremal formally self-dual additive even or odd codes over F. In Section, we give the balance principle and the weak balance principle for formally self-dual additive codes and classify extremal formally self-dual additive odd codes of lengths up to 7 and shows that there is no extremal formally self-dual additive odd code of length n 8. In particular, we construct exactly three formally self-dual additive (7, 7 ) odd codes over F with minimum distance d =, a better minimum distance than any additive self-dual (7, 7 ) codes over F. These (7, 7, ) additive codes over F would produce binary [8, 1, 7] codes or optimal binary [8, 1, 8] codes via Construction O or Construction E respectively, as described in (Kim et.al., 00). In Section, we describe possible weight enumerators of formally self-dual additive odd codes with even lengths up to 1. Our results are F -analogues of binary near-extremal formally self-dual codes considered in (Kim and Pless, 007). We show that there exist near-extremal formally self-dual additive codes of length 6 with all possible weight enumerators. We summarize our results and suggest some open problems in the last section.

3 . Preliminaries We refer to (Calderbank et. al., 1998), (Gaborit, et. al., 001), (Huffman and Pless, 00) for definitions and facts about additive codes over F. All codes considered in this paper mean codes over F unless specified. An additive code C of length n over F is an additive subgroup of F n. C contains k codewords for some 0 k n, and can be defined by a k n generator matrix, with entries from F, whose rows span C additively. We call C an (n, k ) code. We denote F = {0, 1, ω, ω }, where ω = ω + 1. The conjugation of x F is defined by x = x. The trace map, T r : F F, is defined by T r(x) = x + x. The trace inner product of two vectors u = (u 1, u,..., u n ) and v = (v 1, v,..., v n ) in F n is given by n n u, v = T r(u v) = T r(u i v i ) = (u i vi + u i v i ) (mod ). i=1 We define the dual of the code C with respect to the trace inner product by C = {u F n u, c = 0 for all c C}. Then C is also additive. C is called self-orthogonal if C C. If C = C, then C is called self-dual and must be an (n, n ) code. A self-dual code C is Type II if all codewords have even weight and Type I if there is a codeword of odd weight. Two additive codes C 1 and C are equivalent if there is a map sending the codewords of C 1 onto the codewords of C where the map consists of a permutation of coordinates followed by a possible scaling of coordinates by nonzero elements of F followed by possible conjugation of some of the coordinates. The automorphism group Aut(C) of C is the group of all maps sending C to itself using these three operations. The Hamming weight of u, denoted wt(u), is the number of nonzero components of u. The Hamming distance between u and v is wt(u v). The minimum distance of the code C is the minimal Hamming distance between any two distinct codewords of C. Since C is an additive code, the minimum distance is also given by the smallest nonzero weight of any codeword in C. An additive code with minimum distance d is called an (n, k, d) code. The weight distribution of the code C is the sequence (A 0, A 1,..., A n ), where A i is the number of codewords of weight i. The weight enumerator of C is the polynomial n W C (x, y) = A i x n i y i. i=0 i=1 The following fact is well known (see e.g. (Höhn, 00)). Fact (MacWilliams identity) Let C be an additive code over F. Then W C (x, y) = 1 C W C(x + y, x y). Definition 1. An additive code C over F is formally self-dual (f.s.d.) if W C (x, y) = W C (x, y). An additive code C over F is called isodual if C is equivalent to its dual. Clearly isodual additive codes are formally self-dual. A formally self-dual additive (f.s.d.a.) code C over F is even if all the weights of codewords of C are divisible by, and odd if some of the weights of codewords of C are not divisible by.

4 Proposition. Let C be an f.s.d.a. (n, n ) code over F. Then the weight enumerator W C (x, y) is a polynomial in x + y and y(x y). Proof. The proof is essentially the same as the one in Theorem of (Höhn, 00). Proposition. An f.s.d.a. even code C is a self-dual code (of Type II). Proof. Let u, v C. The proposition follows from the following identity. wt(u + v) wt(u) wt(v) u, v (mod ) Let C be an (n, n, d) f.s.d.a. code over F. Define m = n/. By Proposition the weight enumerator of C can be written as m W C (x, y) = a i (x + y) n i (y(x y)) i (1) i=0 with unique constants a i. There is a unique choice of the numbers a 0,..., a m such that the right hand side of (1) equals x n + 0 x n 1 y x n m y m + A m+1 x n m 1 y m A n y n. () We call () the extremal weight enumerator and a code with this extremal weight enumerator has minimum distance d n/ + 1. On the other hand, we have the following. Proposition. The minimum distance d of an f.s.d.a. code C over F satisfies n d + 1. of length n Proof. The proof is essentially the same as the one in Theorem 11 of (Höhn, 00). Hence it is natural to consider the following definitions. Definition. An f.s.d.a. odd code over F of length n is called extremal if its minimum distance is d = n + 1, and near-extremal if its minimum distance is d = n.. Construction of extremal formally self-dual additive odd codes over F By Proposition, it is natural to study f.s.d.a. odd codes over F. In this section we classify (n, n, n +1) extremal f.s.d.a. odd codes. We consider the following construction method, which is a modified balance principle for self-dual codes over F (Gaborit, et. al., 001) and for binary formally self-dual codes (Fields, et. al., 1999), (Huffman and Pless, 00). Let {X} denote the additive code with generator matrix X, k X the F -dimension of {X}, and the matrix O the all-zero matrix of a proper size.

5 Theorem 6. (Balance Principle) Let C and C be the additive codes of length n 1 + n with the following generator matrices G(C) and G(C ), under the assumption that A (respectively B) and F (respectively J) generate the subcodes of C and C of largest dimension with support n 1 (respectively n ): A O F O G(C) = O B and G(C ) = O J D E L M Suppose that dim C =dim C. Then the following hold. (i) k D = k E = k L = k M (ii) {A} = {F L}, {B} = {J M}, {F } = {A D}, {J} = {B E} (iii) n 1 k F = n k B and n 1 k A = n k J Proof. The proofs of (i) and (ii) are exactly the same as in (Fields, et. al., 1999). For (iii) note that n 1 = k A +k F +k L and n = k B +k J +k M = k B +k J +k L by (i) and (ii). Then n 1 (k A + k F ) = n (k B + k J ) = n (k B + k A + k B k F ), where the second equality follows from k A + k B = k F + k J by (i). This implies that n 1 k F = n k B. Similarly we have n 1 k A = n k J. We note that any of k A, k B, k F, k J can be zero in Theorem 6 if A, B, F, or J does not generate the desired subcode. For example, when n = in the below classification, G(C,1 ) has k A = 1 and k B = 0. In fact, by absorbing [A O] into [D E] in G(C) in Theorem 6, we can simplify the computation of the classification of extremal or nearextremal f.s.d.a. codes because D can be given explicitly. To show this we consider the following weak balance principle. Theorem 7. (Weak Balance Principle) Let C and C be the additive codes of length n 1 + n. Suppose that C has a generator matrix F O G(C ) = O J, L M where F (respectively J) is the subcode of C of largest dimension with support n 1 (respectively n ), then the generator matrix G(C) of C can be chosen as G(C) = O B, D E where B generates the subcode of C of largest dimension with support n. Suppose further that dim C =dim C. Then the following hold. (i) k L = k M (ii) {F } = {D} and {J} = {B E} (iii) n 1 k F = n k B

6 Proof. As F (respectively J) is the subcode of C of largest dimension with support n 1 (respectively n ), it follows that k L = k M. This proves (i). It is easy to see (ii). For (iii), we note that n 1 = k F +k D by the first claim of (ii). Since dim C =dim C = n 1 +n, we have n 1 +n = k B +k D. Hence n 1 k F = n 1 +n k B if and only if n 1 k F = n k B. The weak balance principle implies the following two corollaries. Corollary 8. Let C be an (n, n ) extremal f.s.d.a. odd code with minimum distance d = n + 1 and G be its n by n generator matrix. Assume that n is odd. Then G is equivalent to G shown below. G = D 1 E 1 D E where D 1 is the d by d identity matrix, D is the (d 1) by d matrix of the form ω ω ω 0 ω 0 0 D =,. ω ω and E 1 (respectively E ) is an F -matrix of size d (d 1) (respectively (d 1) (d 1)). Further up to equivalence, we can choose the by d submatrix formed by the first two rows of E 1 to be one of the following forms or 1 1 1, () 1 ω ω ω ω ω Proof. We use the above notations. Let n 1 = d. As n = d 1, we know k B = 0. Since C is a formally self-dual additive code with minimum weight d, there exists a codeword c of weight d in C. We may assume that c is the all one vector up to equivalence, and its support is placed in the first d positions. This gives k F 1. Suppose that k F = 1, that is, F is the all one vector of length d. Note that D 1 and D satisfies that {D 1 D } = {F }. Hence we can choose D in Theorem 7 to be D 1 D. Next suppose that k F =, that is, F is generated by (1,..., 1) and (w,..., w) of length d up to equivalence. Then by (iii) of Theorem 7 we have d = (d 1) k B, whence k B = 1. This contradicts with k B = 0. Clearly k F is not possible due to the weight restriction on C. The last statement can be easily proved by considering the minimum distance and the equivalence. Similarly one can show the following when n is even and d is n/. (Note: As seen below, there exist extremal f.s.d.a. odd codes only at code length if the code length n is even. We omit a similar corollary for this case.) 6

7 Corollary 9. Let C be an (n, n ) f.s.d.a. odd code with minimum distance d = n and G be its n by n generator matrix. Assume that n is even. Then G is equivalent to one of the following two matrices G 1 = D 1 E 1 or G 0 0 w w = D wd E E wd E where D 1 is the d by d identity matrix, D is the (d 1) by d matrix of the form D =., E 1 is an F -matrix of size d d, and E, E, E are F -matrices of size (d 1) d. Proof. The proof is similar to that of Corollary 8. Let n 1 = d. In this corollary, we have two possibilities, namely, k F = 1 and k F =. If k F = 1 (respectively k F = ) then k B = 1 (respectively k B = ) by (iii) of Theorem 7. The remaining part of the proof is omitted since it similar to that in Corollary 8. The most time consuming part of Corollary 8 and Corollary 9 is to fill in the entries of E i, (1 i ). We do this by Magma (Canon and Playouts, 199) using the equivalence of additive codes developed in (Gaborit, et. al., 001). We have the following result. In this section, W C (1, y) denotes the extremal weight enumerator. n = 1 : W C (1, y) = 1+y : There is a unique (1,, 1) f.s.d.a. code with generator matrix [1]. This is self-dual. n = : W C (1, y) = 1 + y : There is no extremal f.s.d.a. odd code of length. Only one extremal f.s.d.a. even code generated by (1, 1) and (ω, ω) exists (Höhn, 00). This is a Type II self-dual code. It is easy to check by hand that there are, up to equivalence, exactly two (,, 1) f.s.d.a. non self-dual codes, generated by {(1, 0), (ω, 0)} or {(1, 0), (ω, 1)}, respectively. It is easy to check that both codes are isodual. n = : W C (1, y) = 1 + y + y : We show that there are exactly two extremal f.s.d.a. codes of length, denoted by C,1 and C,. We have the following generator matrices respectively using Corollary G(C,1 ) = , G(C,) = 0 1 ω. ω ω ω ω ω ω ω ω 1 We note that C,1 is a self-dual code (Höhn, 00), while C, is not a self-dual code. C, is generated by {(1, ω, 1), (ω, 0, ω), (0, ω, 1)}. It turns out that C, is isodual. 7

8 Furthermore, we check that Aut(C,1 ) = and Aut(C, ) = 6 (see (Calderbank et. al., 1998) or (Gaborit, et. al., 001) for how to find the automorphism group of an additive code). n = : W C (1, y) = 1 + 1y + y : There is no (,, ) additive self-dual code (Höhn, 00). We apply the weak balance principle (Theorem 7). Let n 1 =. Clearly k B = 0 since the size of support of B is 1. Then by (iii) of Theorem 7, we have k F =. We may assume that F is generated by 111 and ωωω. So {D} = {F } is generated by {(1, 1, 0), (1, 0, 1), (ω, ω, 0), (ω, 0, ω)}. Therefore we have a unique extremal f.s.d.a. code C of length, which has Aut(C ) = 7, and whose generator matrix is unique up to equivalence as shown below ω G(C ) =. ω ω 0 ω ω 0 ω ω Therefore n = is the first length for which a f.s.d.a. code has a better minimum distance than any self-dual additive code over F of that length. The code C is also a linear code over F generated by (1, 1, 0, 1) and (1, 0, 1, ω). Note that C regarded as a linear code over F is not self-dual with respect the usual (Euclidean) inner product. We can further choose a Euclidean self-dual code over F as follows. Let C be a linear code over F with the following generator matrix ω ω Then by Section. in (Rains and Sloane, 1998), C is a Euclidean self-dual code with the weight enumerator W C (1, y) = 1 + 1y + y. Since the MacWilliams identity of a Euclidean self-dual code over F is the same as that of an f.s.d.a. code over F, C is a (,, ) f.s.d.a. code over F. It is straightforward to check that C is equivalent to C as an additive code. n = : W C (1, y) = y + 1y + 6y : Using Corollary 8, we show that there are exactly four (,, ) f.s.d.a. (non self-dual) codes, denoted by C,1,..., C, and a unique (,, ) Type I self-dual code C,. Their generator matrices G(C,i ) for i = 1,..., are given below, and their automorphism group orders are,, 8, 1, 10, respectively. Furthermore, the four non self-dual codes C,1,..., C, are isodual. We note that C, must be equivalent to the unique (,, ) Type I self-dual code (Höhn, 00) ω ω G(C,1) = ω 1, G(C,) = ω 1, 6 ω ω 0 ω ω ω 0 ω 1 7 ω 0 ω 1 ω ω 0 ω ω ω 8

9 ω ω G(C, ) = ω 1, G(C, ) = ω ω, 6 ω ω 0 ω ω ω 0 ω 1 7 ω 0 ω 1 ω ω 0 ω 1 ω ω ω G(C, ) = ω ω. 6 ω ω 0 ω 1 7 ω 0 ω 1 ω n = 6 : W C (1, y) = 1 + y + 18y 6 : Due to this weight enumerator, there is no extremal f.s.d.a. odd code over F. It is known that there is a unique Type II self-dual code of length 6 (Höhn, 00). Let us consider G 1 of Corollary 9. Since the minimum weight is, we can choose the first row of E 1 in G 1 as (0, ω, ω), (ω, ω, ω), or (0, 1, ω) up to equivalence. Note we do not need (1, ω, ω) since after adding (1, 1, 1) to it one will get (0, ω, ω ), which then will be equivalent to (0, ω, ω). Hence using G 1 and G of Corollary 9, we construct exactly 9 (6, 6 ) f.s.d.a. odd non self-dual codes with d =, exactly 1 of which are isodual, and the unique (6, 6, ) Type I self-dual code (Höhn, 00) up to equivalence. In this calculation we have used the restricted equivalence described in (Gaborit, et. al., 001). We have also double checked this result in a brute force way. All the (6, 6, ) codes are posted on the website of the second author. n = 7 : W C (1, y) = 1 + y + y + 8y 6 + y 7 : Each case in Equation () of Corollary 8 produces exactly three inequivalent (7, 7, ) f.s.d.a. (non self-dual) codes. As the three codes of the first case of Corollary 8, denoted by C 7,1,..., C 7,, are equivalent to those of the second case, we only display their generator matrices below, and their automorphism group orders are 7, 6,, respectively. Further, it is checked that these three codes are isodual. There is no (7, 7, ) Type I self-dual code but there exist four (7, 7, ) Type I self-dual codes (note: the three codes with these parameters in Table 1 of (Gaborit, et. al., 001) are corrected in (Danielsen and Parker, 006)). Thus just like the n = case, the minimum distance of extremal f.s.d.a. codes of length n = 7 beats that of any self-dual codes of the same length. Hence applying Construction O or Construction E (Kim et.al., 00) to the three extremal f.s.d.a. codes we get binary [8, 1, 7] codes or optimal binary [8, 1, 8] codes (Brouwer, 1998). 9

10 ω ω ω ω ω ω ω ω 1 G(C 7,1) = ω 1 ω, G(C 7,) = ω 1 ω, ω ω 0 0 ω ω 1 ω ω 0 0 ω ω ω 6 ω 0 ω 0 ω ω ω 7 6 ω 0 ω 0 ω ω ω 7 ω 0 0 ω 1 ω ω ω 0 0 ω ω ω ω ω ω ω ω 1 G(C 7, ) = ω ω ω. ω ω 0 0 ω ω ω 6 ω 0 ω 0 ω 1 ω 7 ω 0 0 ω ω ω 1 It was remarked by a reviewer that Blockhuis and Brouwer have already found one (7, 7, ) additive code over F (see Blokhuis et. al. (00)). Further, an alternative geometric construction of this particular code has been described by Bierbrauer et. al. (see Bierbrauer et. al. (006)). We have checked that this code is equivalent to our C 7,. However the f.s.d.a. nature of this code was not mentioned in Blokhuis et. al. (00) nor were the other two (7, 7, ) f.s.d.a. codes found in these two papers. n = 8 : negative weight enumerator (A n + < 0). Hence there is no extremal f.s.d.a. code of length 8. n = 9 : W C (1, y) = 1+16y +8y y y 8 +y 9 : Using Corollary 8, we have checked that there is no extremal f.s.d.a. code of length 9. On the other hand, using a similar form as in Corollary 8, we have found at least 178 (9, 9, ) near-extremal f.s.d.a. codes, all of which are neither self-dual nor isodual. We only give different weight enumerators of these codes as follows. W 9 (1, y) = 1 + (17 + a)y + (7 a)y + (118 + a)y 6 + (1 + a)y 7 + (10 a)y 8 + (9 + a)y 9, where a = 0, 1,,,,, 6, 7, and 8. For example, a (9, 9, ) f.s.d.a. code with W 9 (1, y) and a = 0 has generator matrix G(C 9,1 ) whose rows are generated by the following vectors: {(1, 0, 0, 0, 0, 1, 1, 1, 1), (0, 1, 0, 0, 0, 1, ω, ω, ω), (0, 0, 1, 0, 0, 1, ω, 1, 0), (0, 0, 0, 1, 0, ω, ω, 1, 0), (0, 0, 0, 0, 1, ω, 1, 1, 0), (ω, ω, 0, 0, 0, ω, ω, 1, 0), (ω, 0, ω, 0, 0, 1, 0, ω, 1), (ω, 0, 0, ω, 0, ω, 0, 1, ω), (ω, 0, 0, 0, ω, 1, ω, ω, ω )}. Other generator matrices are posted on the website of the second author. n = 10 : negative weight enumerator (A n + < 0). Hence there is no extremal f.s.d.a. code of length 10. n = 11 : W C (1, y) = 1+6y 6 +0y 7 +9y y 9 +66y 10 +1y 11 : Using Corollary 8, we have checked that there is no extremal f.s.d.a. code of length

11 Table 1. Highest minimum distance of formally self-dual additive odd (f.s.d.a.o) non self-dual codes over F of lengths up to 1 length non sd dfsdao non sd numfsdao d sd,i num sd,i 1 NE 1 1 E 1 1 E 1 1 E 1 6 NE E 8 NE 10 [GO] 9 NE NE [GO] NE? NE 1 [GO] NE? or 7 NE 9 [GO] or? 6 n 1 : A n + < 0 by the proof of Theorem 1 in (Höhn, 00). Hence there is no extremal f.s.d.a. code of length n if n 1. non sd The above results are summarized in Table 1. Here the second column dfsdao refers to the (extremal (E) or near-extremal (NE)) minimum distance of possible formally self-dual additive odd codes excluding Type I self-dual codes, the third column refers to the number of corresponding codes where GO refers to the reference (Gulliver and Östergård, 00), and the fourth and fifth columns refer to the minimum distance of optimal Type I self-dual codes and the number of the corresponding codes respectively from (Danielsen and Parker, 006), (Gaborit, et. al., 001), (Höhn, 00), (Huffman, 00), and (Varbanov, 007). In particular, we emphasize the following. Theorem 10. (i) There exists a unique (,, ) extremal f.s.d.a. code over F. Its minimum distance is higher than any (, ) additive self-dual code over F. (ii) There are exactly four (,, ) extremal f.s.d.a. odd non self-dual codes over F. (iii) There are exactly 9 (6, 6, ) near-extremal f.s.d.a. odd non self-dual codes over F. (iv) There are exactly three (7, 7, ) extremal f.s.d.a. codes over F. Their minimum distance is higher than any (7, 7 ) additive self-dual code over F. (v) Any (n, n, d) f.s.d.a. odd code over F satisfies d n for n 8. 11

12 . Weight enumerators of near-extremal f.s.d.a. odd codes over F of even lengths 1 In this section we calculate the possible weight enumerators of near-extremal f.s.d.a. odd codes of even lengths up to 1. Our results are F -analogues of binary near-extremal formally self-dual codes done in (Kim and Pless, 007). Let C be an f.s.d.a. odd code over F. We define a codeword in C to be even if its weight is even and odd if its weight is odd. We denote the set of even codewords in C by EC and the set of odd codewords in C by OC. We call an f.s.d.a. odd code balanced if it contains an equal number of even codewords and odd codewords. By Proposition we have W C (x, y) = n/ k=0 for some a k. It follows from W C (1, y) = n i=0 A iy i that Thus combining () and (), we have the following. a k (x + y) n k (y(x y)) k () W C (1, 1) = EC OC. () Proposition 11. If C is an f.s.d.a. odd code over F of odd length, then C is balanced. Let C be a near-extremal f.s.d.a. odd code over F of even length n. Then the coefficients a 0, a 1,..., a (n/) 1 in () are uniquely determined. We denote the coefficient a n in () as α. Then W C (1, 1) = α( ) n n n = ( 1) α. So we have This implies the following result. EC = OC if and only if α = 0. (6) Proposition 1. The weight distribution of a near-extremal f.s.d.a. odd code over F of even length and EC = OC is unique, and is given by () with α (= a n ) = 0. Now we want to calculate the possible values of α for a non-balanced near-extremal f.s.d.a. odd code over F of even length. Before we do that, we need the following results which are stated in (Pless, 1997) (or see Sec. 7.8 of (Huffman and Pless, 00)). A binary linear code is called even if it only contains even weight vectors. A doublyeven (d.e.) vector has weight 0 (mod ), while a singly-even (s.e.) vector has weight (mod ). A hyperbolic plane is a two dimensional space generated by two doublyeven vectors which are not orthogonal to each other. An anisotropic plane is generated by two singly-even non-orthogonal vectors. We write C 1 C to mean the vector space direct sum of two codes C 1 and C which are orthogonal to each other. If C is an even binary code, let R(C) denote the largest doubly-even subcode of C C and let r = dimr(c). Let a denote the number of d.e. vectors in C and b denote the number of s.e. vectors in C. Then every even binary linear [n, k] code C is one of three types (see (Pless, 1997) or (Huffman and Pless, 00, Ch. 7)). (i) Hyperbolic Type. Here C = R(C) H m where H m is the orthogonal sum of m hyperbolic planes. Clearly k = r + m. In this case the following holds 1

13 a = r ( m 1 + m 1 ), b = r ( m 1 m 1 ). (ii) Anisotropic Type. Here C = R(C) H (m 1) A where H (m 1) is the orthogonal sum of (m 1) hyperbolic planes and A is an anisotropic plane. Again k = r + m. Further, a = r ( m 1 m 1 ), b = r ( m 1 + m 1 ). (iii) Odd Anisotropic Type. Here C = R(C) H m < x > where x is a singly-even vector. Now k = r + m + 1 and a = b = k 1. Now we are ready to prove the following theorem. Theorem 1. Let C be an (n, n, n ) near-extremal f.s.d.a. odd code over F of even length. Then the possible coefficient α (= a n ) in () is given by Furthermore, EC and OC are given by EC = n 1 + α n 1, OC = n 1 α n 1. α = 0 or ± i for i = 0, 1,,..., n 1. Proof. If α = 0, then the theorem holds. So we assume that α 0. Define φ : F F by φ(0) = (0, 0, 0), φ(1) = (1, 1, 0), φ(ω) = (1, 0, 1), φ(ω) = (0, 1, 1). Define φ n : F n F n by φ n (a 1, a,..., a n ) = (φ(a 1 ), φ(a ),..., φ(a n )). Then φ n is F -linear, and φ n (C) is a [n, n, n ] binary linear even code. Let a be the number of doubly-even codewords in φ n (C) and b be the number of singly-even codewords in φ n (C). Then we have EC = a and OC = b. As α 0, we have a b, and the Odd Anisotropic case does not occur. Using the notations before Theorem 1, we have n = r + m, a b = ± r+m = ( 1) n α n, a + b = n. 1

14 Hence r is even as n is even, and a = n 1 + ( 1) n α n 1 and b = n 1 ( 1) n α n 1. As n = r + m, ( α = ±( 1) n r r, = 0, 1,..., n ). Now we only have to prove that r n. Suppose r = n. Then φ n(c) = R(φ n (C)). This is impossible since C is an odd additive code over F. Now we state possible weight enumerators W (1, y) with α (= a n ) for small code lengths. n = 6 : W (1, y) = 1 + (8 + α)y + (1 α)y + ( + α)y + (10 α)y 6. The 9 f.s.d.a. odd codes over F of length 6 with d = in Section produce all possible values of α =,, 1, 0, 1,,. We give only seven codes with each α from to, denoted by C 6,1, C 6,,..., C 6,7, respectively. Their generator matrices are given in Table. n = 8 : W (1, y) = 1 + (6 + α)y + (6 α)y + (7 + 6α)y 6 + (6 α)y 7 + (9 + α)y 8. It is shown (Gulliver and Östergård, 00) that there exist near-extremal formally self-dual linear odd codes over F with α = 8,, 1,. n = 10 : W (1, y) = 1+(9+α)y +(170 α)y 6 +(00+10α)y 7 +(9 10α)y 8 +(0+α)y 9 + (6 α)y 10. It is shown (Gulliver and Östergård, 00) that there exist near-extremal formally self-dual linear odd codes over F with α =, 1,. n = 1 : W (1, y) = 1 + ( + α)y 6 + (8 6α)y 7 + ( + 1α)y 8 + (180 0α)y 9 + ( α)y 10 +(8 6α)y 11 +(170+α)y 1. It is shown (Gulliver and Östergård, 00) that there exist near-extremal formally self-dual linear odd codes over F with α =. n = 1 : W (1, y) = 1+(10+α)y 7 +(69 7α)y 8 +(196+1α)y 9 +(8 α)y 10 +(88+ α)y 11 + (6 1α)y 1 + ( α)y 1 + (18 α)y 1. No near-extremal f.s.d.a. odd code over F of this length is known. We want to remark that in (Han and Kim, 009) we show the following. Let C be an (n, n, d) f.s.d.a. odd code over F. If n = 16, 18 or n 0, then d < n. In other words, there is no near-extremal f.s.d.a. odd code over F if n = 16, 18 or n 0.. Conclusion In this paper, we have introduced the class of formally self-dual additive codes over F, defined extremal and near-extremal formally self-dual additive codes over F, classified all extremal codes, and constructed a lot of near-extremal codes. We have discussed several construction methods in detail. We have also given possible weight enumerators of near-extremal f.s.d.a. odd codes over F with even lengths 1. As future work, as one can see in the weak balance principle (Theorem 7) of Section, it will be very interesting to find an algorithm to reduce the time to compute the E part. In general, find other methods to construct formally self-dual additive codes over F or isodual additive codes over F. 1

15 Table. Near-extremal formally self-dual additive odd codes over F of length 6 with α =,, 1, 0, 1,,, respectively (only 7 codes are shown) w w w w w w w w 0 G(C 6,1) = w w w, G(C 6,) = w w w, 6 w w w w w 0 w 1 w 0 w 0 w 1 w w w w w w w w w 0 G(C 6, ) = w w w, G(C 6, ) = w w w, 6 w w w w w 0 w w 0 0 w 0 w w w w w w w w w w w G(C 6,) = w w w, G(C 6,6) = w w w w, 6 w w w w 0 w w 0 w w 0 w w w w w w 1 0 G(C 6,7 ) = w w. 6 w w 0 w w 0 7 w 0 w w 0 0 Acknowledgements Most of this work was done while S. Han was visiting the Department of Mathematics at the University of Louisville. S. Han would like to thank the university. J.-L. Kim would like to thank P. Solé for mentioning his paper (Choie and Solé, 008). We also thank the referees for valuable comments. 1

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