09-05-2013, 03:02 PM
Note on depth spectrum and depth distribution of binary code
ABSTRACT
It is shown that the depth distribution of a linear binary code can
be determined if its depth spectrum is given. Then the relation
between the depth spectra for a linear binary code and its dual
code as well as that for a linear binary code and its extended code
are considered, and some results given.
Introduction:
The notion of the depth of a finite binary sequence
was first introduced by Etzion [l]. Etzion showed that a linear
code of dimension k contains codewords of k distinct depths, and
also gave the depth spectra for the Hamming code, the extended
Hamming code, and the first-order Reed-Muller code. Mitchell [2]
extended the notion of the depth of a finite binary sequence to an
infinite binary sequence, and showed that the set of infinite
sequences of finte depth corresponds to a set of equivalence
classes of rational polynomials. For all linear cyclic codes, Mitchell
found the depth spectra.
This Letter is a note on the work in [l, 21. It is shown that the
depth distribution of a linear binary code can be determined if its
depth spectra are given. The relation between the depth spectra for
a linear binary code and its dual code as well as that for a linear
binary code and its extended code are then considered, and some
results given.