Preparing
of calculations in GF(q) of order two extension for BCH-code.
V. P. Poltorak
National Technical University of Ukraine “Kyiv
Polytechnic Institute”, Informatics and Computer Engineering faculty, 03056, Kyiv, Peremogy avenue, 37,
Ukraine, poltorak@acts.kiev.ua
It is known a renewal of interest to cyclic Error Correcting Codes for
some engineering branches at resent years. It is true for those branches where
we can not accept an Error Detection only and wait for an Error Correcting by
means of data retransmitting. We
should to correct errors in real-time data flow. In some of such engineering
branches BCH-codes are used [1]. For example, at real-time video-conferencing
over packet-switched networks.
From the other hand, a general definition of the BCH-code not guarantee
a code redundancy D = r/n minimization, where n is code length and r =
2mt is a number of code redundant elements, t is a number of errors corrected per code block, m
is an order of GF(q) extension up to GF(qm), q = pl
is a code base (an integer power l of prime number p), GF(q) is
Galois Field of order q [2]. We should to look for a certain code parameter sets for the code redundancy D
minimization. It is well known now, that increasing of q lead to the
code redundancy reducing
even if a short code length n. That is why, we are interested in non
binary BCH-codes (with base q > 2). GF(q) is known as field of code
elements and GF(qm) – as field of locators.
It was found, in special case when m = 2, a set of roots of code
generating polynomial g(x) can be choose in a such way, that we get a
value of r = 2t, even if an m =2 (a general BCH-code
theory promise such possibility, but not show the way to get the result). And
we get a value of D =2t/n - the
least from existing. That fact open attractive perspective in BCH-code using
thanks to D minimization.
Decoding procedure needs of calculations over GF(qm). It is
much more complex, then calculations over GF(q). In special case, presented
above, at m = 2, was obtained a
possibility to reduce a calculations complexity over GF(q2) thanks
to a special form of GF(q2) elements presentation and taking to
consideration of their connection with GF(q) elements.
References
1. S. Lin and D. Costello. Error Control Coding: Fundamentals and
Applications. Prentice-Hall,
2. http://en.wikipedia.org/wiki/Finite_field