28 research outputs found
The correspondence between projective codes and 2-weight codes
The hyperplanes intersecting a 2-weight code in the same number of points obviously form the point set of a projective code. On the other hand, if we have a projective code C, then we can make a 2-weight code by taking the multiset of points <c >E PC with multiplicity "Y(w), where W is the weight of c E C and "Y( w) = aw + f3 for some rational a and f3 depending on the weight enumerator of C. In this way we find a 1-1 correspondence between projective codes and 2-weight codes. The second construction can be generalized by taking for "Y{ w) a polynomial of higher degree. In that case more information about the cosets of the dual of C is needed. Several new ternary codes will be constructed in this way
The weight distribution and randomness of linear codes
Finding the weight distributions of block codes is a problem of theoretical and practical interest. Yet the weight distributions of most block codes are still unknown except for a few classes of block codes. Here, by using the inclusion and exclusion principle, an explicit formula is derived which enumerates the complete weight distribution of an (n,k,d) linear code using a partially known weight distribution. This expression is analogous to the Pless power-moment identities - a system of equations relating the weight distribution of a linear code to the weight distribution of its dual code. Also, an approximate formula for the weight distribution of most linear (n,k,d) codes is derived. It is shown that for a given linear (n,k,d) code over GF(q), the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q = q(-)(n-k) as u becomes large. A relationship between the randomness of a linear block code and the minimum distance of its dual code is given, and it is shown that most linear block codes with rigid algebraic and combinatorial structure also display certain random properties which make them similar to random codes with no structure at all
On the Golay perfect binary code
AbstractSome combinatorial properties of the Golay binary code are emphasized and used for two main purposes. First, a new nonlinear code having 256 code words of length 16 at mutual distance 6 is exhibited. Second, a majority decoding method, quite similar to Massey's threshold decoding, is devised for the Golay code and various related codes
Blocked regular fractional factorial designs with minimum aberration
This paper considers the construction of minimum aberration (MA) blocked
factorial designs. Based on coding theory, the concept of minimum moment
aberration due to Xu [Statist. Sinica 13 (2003) 691--708] for unblocked designs
is extended to blocked designs. The coding theory approach studies designs in a
row-wise fashion and therefore links blocked designs with nonregular and
supersaturated designs. A lower bound on blocked wordlength pattern is
established. It is shown that a blocked design has MA if it originates from an
unblocked MA design and achieves the lower bound. It is also shown that a
regular design can be partitioned into maximal blocks if and only if it
contains a row without zeros. Sufficient conditions are given for constructing
MA blocked designs from unblocked MA designs. The theory is then applied to
construct MA blocked designs for all 32 runs, 64 runs up to 32 factors, and all
81 runs with respect to four combined wordlength patterns.Comment: Published at http://dx.doi.org/10.1214/009053606000000777 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A complementary design theory for doubling
Chen and Cheng [Ann. Statist. 34 (2006) 546--558] discussed the method of
doubling for constructing two-level fractional factorial designs. They showed
that for , all minimum aberration designs with runs
and factors are projections of the maximal design with factors
which is constructed by repeatedly doubling the design defined by
. This paper develops a general complementary design theory for
doubling. For any design obtained by repeated doubling, general identities are
established to link the wordlength patterns of each pair of complementary
projection designs. A rule is developed for choosing minimum aberration
projection designs from the maximal design with factors. It is further
shown that for , all minimum aberration designs with
runs and factors are projections of the maximal design with runs and
factors.Comment: Published in at http://dx.doi.org/10.1214/009005360700000712 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org