565 research outputs found
Self-Dual Codes
Self-dual codes are important because many of the best codes known are of
this type and they have a rich mathematical theory. Topics covered in this
survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight
enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems,
bounds, mass formulae, enumeration, extremal codes, open problems. There is a
comprehensive bibliography.Comment: 136 page
The invariants of the Clifford groups
The automorphism group of the Barnes-Wall lattice L_m in dimension 2^m (m not
3) is a subgroup of index 2 in a certain ``Clifford group'' C_m (an
extraspecial group of order 2^(1+2m) extended by an orthogonal group). This
group and its complex analogue CC_m have arisen in recent years in connection
with the construction of orthogonal spreads, Kerdock sets, packings in
Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs.
In this paper we give a simpler proof of Runge's 1996 result that the space
of invariants for C_m of degree 2k is spanned by the complete weight
enumerators of the codes obtained by tensoring binary self-dual codes of length
2k with the field GF(2^m); these are a basis if m >= k-1. We also give new
constructions for L_m and C_m: let M be the Z[sqrt(2)]-lattice with Gram matrix
[2, sqrt(2); sqrt(2), 2]. Then L_m is the rational part of the mth tensor power
of M, and C_m is the automorphism group of this tensor power. Also, if C is a
binary self-dual code not generated by vectors of weight 2, then C_m is
precisely the automorphism group of the complete weight enumerator of the
tensor product of C and GF(2^m). There are analogues of all these results for
the complex group CC_m, with ``doubly-even self-dual code'' instead of
``self-dual code''.Comment: Latex, 24 pages. Many small improvement
On Asymmetric Coverings and Covering Numbers
An asymmetric covering D(n,R) is a collection of special subsets S of an
n-set such that every subset T of the n-set is contained in at least one
special S with |S| - |T| <= R. In this paper we compute the smallest size of
any D(n,1) for n <= 8. We also investigate ``continuous'' and ``banded''
versions of the problem. The latter involves the classical covering numbers
C(n,k,k-1), and we determine the following new values: C(10,5,4) = 51,
C(11,7,6,) =84, C(12,8,7) = 126, C(13,9,8)= 185 and C(14,10,9) = 259. We also
find the number of nonisomorphic minimal covering designs in several cases.Comment: 11 page
The Lattice of N-Run Orthogonal Arrays
If the number of runs in a (mixed-level) orthogonal array of strength 2 is
specified, what numbers of levels and factors are possible? The collection of
possible sets of parameters for orthogonal arrays with N runs has a natural
lattice structure, induced by the ``expansive replacement'' construction
method. In particular the dual atoms in this lattice are the most important
parameter sets, since any other parameter set for an N-run orthogonal array can
be constructed from them. To get a sense for the number of dual atoms, and to
begin to understand the lattice as a function of N, we investigate the height
and the size of the lattice. It is shown that the height is at most [c(N-1)],
where c= 1.4039... and that there is an infinite sequence of values of N for
which this bound is attained. On the other hand, the number of nodes in the
lattice is bounded above by a superpolynomial function of N (and
superpolynomial growth does occur for certain sequences of values of N). Using
a new construction based on ``mixed spreads'', all parameter sets with 64 runs
are determined. Four of these 64-run orthogonal arrays appear to be new.Comment: 28 pages, 4 figure
Quantum Error Correction and Orthogonal Geometry
A group theoretic framework is introduced that simplifies the description of
known quantum error-correcting codes and greatly facilitates the construction
of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1
error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors,
and 1 to 29 qubits correcting 5 errors.Comment: RevTex, 4 pages, no figures, submitted to Phys. Rev. Letters. We have
changed the statement of Theorem 2 to correct it -- we now get worse rates
than we previously claimed for our quantum codes. Minor changes have been
made to the rest of the pape
Quantum Error Correction via Codes over GF(4)
The problem of finding quantum error-correcting codes is transformed into the
problem of finding additive codes over the field GF(4) which are
self-orthogonal with respect to a certain trace inner product. Many new codes
and new bounds are presented, as well as a table of upper and lower bounds on
such codes of length up to 30 qubits.Comment: Latex, 46 pages. To appear in IEEE Transactions on Information
Theory. Replaced Sept. 24, 1996, to correct a number of minor errors.
Replaced Sept. 10, 1997. The second section has been completely rewritten,
and should hopefully be much clearer. We have also added a new section
discussing the developments of the past year. Finally, we again corrected a
number of minor error
- …