A complete classification of ternary self-dual codes of length 24

Ternary self-dual codes have been classified for lengths up to 20. At length 24, a classification of only extremal self-dual codes is known. In this paper, we give a complete classification of ternary self-dual codes of length 24 using the classification of 24-dimensional odd unimodular lattices.     

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over ℤ6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ₆-codes. Then extremal self-dual ₆-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.     

Binary extremal self-dual codes of length 60 and related codes

We give a classification of four-circulant singly even self-dual [60, 30, d] codes for \(d=10\) and 12. These codes are used to construct extremal singly even self-dual [60, 30, 12] codes with weight enumerator for which no extremal singly even self-dual code was previously known to exist. From extremal singly even self-dual [60, 30, 12] codes, we also construct optimal singly even self-dual [58, 29, 10] codes with weight enumerator for which no optimal singly even self-dual code was previously known to exist. Finally, we give some restriction on the possible weight enumerators of certain singly even self-dual codes with shadow of minimum weight 1.     

A bound for certain <Emphasis Type="Italic">s</Emphasis>-extremal lattices and codes

In this paper we introduce the notion of s-extremal lattice for unimodular Type I lattices. We give a bound on the existence of certain such s-extremal lattices: an s-extremal lattice of dimension n and minimal even norm μ must satisfy n < 12μ. This result implies that such lattices are also extremal and that there are a finite number of them. We also give an equivalent bound for s-extremal self-dual codes: an s-extremal code with doubly-even minimum distance d and length n must satisfy n < 6d, moreover such codes are extremal. Received: 25 July 2006     

The binary extremal self-dual codes of lengths 38 and 40

We classify the extremal self-dual codes of lengths 38 or 40 having an automorphism of order 3 with six independent 3-cycles, 10 independent 3-cycles, or 12 independent 3-cycles. In this way we complete the classification of binary extremal self-dual codes of length up to 48 having automorphism of odd prime order.     

Near-Extremal Formally Self-Dual Even Codes of Lengths 24 and 32

The weight enumerator of a formally self-dual even code is obtained by the Gleason theorem. Recently, Kim and Pless gave some restrictions on the possible weight enumerators of near-extremal formally self-dual even codes of length divisible by eight. In this paper, the weight enumerators for which there is a near-extremal formally self-dual even code are completely determined for lengths 24 and 32, by constructing new near-extremal formally self-dual codes. We also give a classification of near- extremal double circulant codes of lengths 24 and 32. Communicated by: P. Fitzpatrick     

There exists no self-dual [24,12,10] code over $${{\mathbb F}_5}$$

Self-dual codes over exist for all even lengths. The smallest length for which the largest minimum weight among self-dual codes has not been determined is 24, and the largest minimum weight is either 9 or 10. In this note, we show that there exists no self-dual [24, 12, 10] code over , using the classification of 24-dimensional odd unimodular lattices due to Borcherds.      

Splitting the shadow

We derive formulae for the theta series of the two translates of the even sublattice L₀ of an odd unimodular lattice L that constitute the shadow of L. The proof rests on special evaluations of the Jacobi theta series attached to L and to a certain vector. We produce an analogous theorem for codes. Additionally, we construct non-linear formally self-dual codes and relate them to lattices.     

An optimal odd unimodular lattice in dimension 72

It is shown that if there is an extremal even unimodular lattice in dimension 72, then there is an optimal odd unimodular lattice in that dimension. Hence, the first example of an optimal odd unimodular lattice in dimension 72 is constructed from the extremal even unimodular lattice which has been recently found by G. Nebe.     

Extremal self-dual codes of length 64 through neighbors and covering radii

We construct extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.     

Classification of Extremal Formally Self-Dual Even Codes of Length 22

 Lengths 22 and 30 are so far the only open cases in the classification of extremal formally self-dual even codes. In this paper, a classification of the extremal formally self-dual even codes of length 22 is given. There are 41520 such codes.A variety of properties of these codes are investigated. In particular, new 2-(22, 6, 5) designs are constructed from the codes. Received: February 9, 2000     

On the Classification of Extremal Even Formally Self-Dual Codes

Bachoc bachoc has recently introduced harmonic polynomials for binary codes. Computing these for extremal even formally self-dual codes of length 12, she found intersection numbers for such codes and showed that there are exactly three inequivalent [12,6,4] even formally self-dual codes, exactly one of which is self-dual. We prove a new theorem which gives a generator matrix for formally self-dual codes. Using the Bachoc polynomials we can obtain the intersection numbers for extremal even formally self-dual codes of length 14. These same numbers can also be obtained from the generator matrix. We show that there are precisely ten inequivalent [14,7,4] even formally self-dual codes, only one of which is self-dual.     

Conformal designs based on vertex operator algebras

We introduce the notion of a conformal design based on a vertex operator algebra. This notation is a natural analog of the notion of block designs or spherical designs when the elements of the design are based on self-orthogonal binary codes or integral lattices, respectively. It is shown that the subspaces of an extremal self-dual vertex operator algebra of fixed degree form conformal 11-, 7-, or 3-designs, generalizing similar results of Assmus and Mattson and Venkov for extremal doubly-even codes and extremal even lattices. Other examples are coming from group actions on vertex operator algebras, the case studied first by Matsuo. The classification of conformal 6- and 8-designs is investigated. Again, our results are analogous to similar results for codes and lattices.     

Construction of Extremal Type II Codes over ℤ4

In this paper, we give a pseudo-random method to construct extremal Type II codes overℤ₄ . As an application, we give a number of new extremal Type II codes of lengths 24, 32 and 40, constructed from some extremal doubly-even self-dual binary codes. The extremal Type II codes of length 24 have the property that the supports of the codewords of Hamming weight 10 form 5−(24,10,36) designs. It is also shown that every extremal doubly-even self-dual binary code of length 32 can be considered as the residual code of an extremal Type II code over ℤ₄.     

Self-dual codes over the Kleinian four group

We introduce self-dual codes over the Kleinian four group K=Z ₂×Z ₂ for a natural quadratic form on K ⁿ and develop the theory. Topics studied are: weight enumerators, mass formulas, classification up to length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs, lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism groups, marked codes. Kleinian codes form a new and natural fourth step in a series of analogies between binary codes, lattices and vertex operator algebras. This analogy will be emphasized and explained in detail.     

Classification of extremal double circulant self-dual codes of lengths 74-88

A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. In this paper, we give a classification of all extremal double circulant self-dual codes of lengths 74-88.     

Classification of Extremal Double Circulant Formally Self-Dual Even Codes

Formally self-dual even codes have recently been studied. Double circulant even codes are a family of such codes and almost all known extremal formally self-dual even codes are of this form. In this paper, we classify all extremal double circulant formally self-dual even codes which are not self-dual. We also investigate the existence of near-extremal formally self-dual even codes.     

The extremal codes of lengths 76 with an automorphism of order 19

We find all extremal [76,38,14] binary self-dual codes having automorphism of order 19. There are three inequivalent such codes. One of them was previously known. The other two are new. These codes are the shortest known self-dual codes of minimal weight 14 as well as the best-known linear codes of that length and dimension.     

A mass formula for unimodular lattices with no roots

We derive a mass formula for -dimensional unimodular lattices having any prescribed root system. We use Katsurada's formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimension . In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimension , verifying Bacher and Venkov's enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants.

     

Classification of Formally Self-Dual Even Codes of Lengths up to 16

In this note, we give the complete classification of binary formally self-dual even codes of lengths 10, 12, 14 and 16. There are exactly fourteen, 29, 99 and 914 inequivalent such codes of lengths 10, 12, 14 and 16, respectively. This completes the classification of formally self-dual even codes of lengths up to 16. The first example of formally self-dual even code with a trivial automorphism group is also found. This shows that 16 is the smallest length for which there is a formally self-dual even code with a trivial automorphism group.