Application of the Combinatorial Configurations for Synthesis of
Correcting Codes
Volodymyr Riznyk, Oleksandr Lesjuk, Marija Ljepikhova
Control Aided Systems Department, Lviv Polytechnic National University, S. Bandery Str., 12, Lviv, 79013,
UKRAINE, E-mail: rvv@polynet.lviv.ua
Abstract – The objective of the paper is application of the
combinatorial configurations for improving quality indices of
coding systems with respect to noise immunity based on the
combinatorial configurations theory. Research into the underlying
mathematical structures relates to the optimal placement of
symbols “0” and “1” in code combinations of the system using
combinatorial techniques. Correcting cyclic codes design using
nonstandard mathematical constructions with special combinative
properties, namely the “Ideal Ring Bundles” (IRB)s are studied.
The techniques for optimum synthesis of the cyclic codes with
improved possibilities for correcting the multiple errors are
regarded.
namely, via an algebraic method known as syndrome
decoding. This simplifies the design of the decoder for these
codes, using small low-power electronic hardware. BCH
codes are used in applications such as satellite
communications, compact disc players, DVDs, disk drives,
solid-state drives and two-dimensional bar codes [2].
Given a prime power q and positive integers m and d with d
≤ qm − 1, a primitive narrow-sense BCH code over the finite
field GF(q) with code length n = qm − 1 and distance at least
d is constructed by the following method [2].
Let α be a primitive element of GF(qm). For any positive
Кеу words – combinatorial configuration, information integer i, let mi(x) be the minimal polynomial of αi over GF(q).
technology, correcting code, error protection, Galois field, coding The generator polynomial of the BCH code is defined as the
system, optimization.
I. Introduction
Transfer of information from one place to another faces many
difficulties. Principal among them is noise. Clearly if what is sent
is not what is received, communication can be problematic. Error
correcting codes have been developed to solve this type of problem.
Computer scientists came up with a simple error detection method
called “parity check”. With this method can be represented data
using only the first 7 bits. The last bit is always chosen so that
together with the other seven there are an even number of 1's in the
byte. When the data arrives at the other end, the receiver counts the
number of 1's in the byte. If it's odd, then the byte must have been
contaminated by noise so the receiver may ask for retransmission.
This method only detects errors but it can not correct them other
than ask for retransmission. If retransmission is expensive (e.g.
satellite), parity check is not ideal. Also its error detection ability is
very low. If two bits were changed by noise, then the receiver will
assume the message is correct. More sophisticated error correction
codes address these problems. No error correcting code is perfect
(although we call some perfect codes). No code can correct every
possible error vector. But it is also reasonable to assume that only a
small number of errors are made each transmission and so we only
need codes that can correct a small number of errors [1].
II.Bose-Chaudhuri-Hocquengrem Codes
In coding theory, the Bose-Chaudhuri-Hocquengrem
codes (BCH codes) form a class of cyclic error-correcting
codes that are constructed using finite fields [2]. BCH codes
were invented in 1959 by French mathematician Alexis
Hocquenghem, and independently in 1960 by Raj Bose and
D. K. Ray-Chaudhuri. The acronym BCH comprises the
initials of these inventors' names. One of the key features of
BCH codes is that during code design, there is a precise
control over the number of symbol errors correctable by the
code. In particular, it is possible to design binary BCH codes
that can correct multiple bit errors. Another advantage of
BCH codes is the ease with which they can be decoded,
least common multiple g(x) = (m1(x),…,md − 1(x)). It can be
seen that g(x) is a polynomial with coefficients in GF(q) and
divides xn − 1. Therefore, the polynomial code defined by g(x)
is a cyclic code. There are many algorithms for decoding BCH
codes. The most common ones follow this general outline:
1.
2.
3.
4.
5.
Calculate the syndromes sj for the received vector
Determine the number of errors t and the error locator
polynomial Λ(x) from the syndromes
Calculate the roots of the error location polynomial to
find the error locations Xi
Calculate the error values Yi at those error locations
Correct the errors
During some of these steps, the decoding algorithm may
determine that the received vector has too many errors and cannot be
corrected. For example, if an appropriate value of t is not found, then
the correction would fail. In a truncated (not primitive) code, an error
location may be out of range. If the received vector has more errors
than the code can correct, the decoder may unknowingly produce an
apparently valid message that is not the one that was sent [2].
III. Ideal Ring Bundle Codes
The Ideal Ring Bundle (IRB) is cyclic sequences of integers,
which form perfect partitions of a finite interval [1, S] of
integers. The sums of connected sub-sequences of the IRB
enumerate the set of integers [1, S] exactly R-times [3].
Underlying combinatorial structure can be represented in
terminology of the theory of finite fields [4]. For any prime
power q these exists a finite field with exactly q elements. This
field is unique up to isomorphism and is called the Galois field
of q elements [written GF(q)]. The multiplicative group of
GF(q) is cyclic; thus it is generated by any of its elements of
order q-1. These generating elements are primitive roots and if
α is a primitive root so is αx whenever x is prime to q-1. For
prime p, the residues 0,1,…,p-1 form a field with respect to
addition and multiplication modulo p; this field is often taken to
be the generic representation of GP(p). GF(r), r=q m, can be
“COMPUTER SCIENCE & INFORMATION TECHNOLOGIES” (CSIT’2014), 18-22 NOVEMBER 2014, LVIV, UKRAINE
constructed from GF(q) by adjoining any root of any m-th
degree polynomial f(x) irreducible over GF(q).
GF(qm) can be represented by the set of all m-tuples with
entries from GF(q). For multiplicative purposes it is more
convenient to represent GF(qm) in terms of a primitive root α;
in which case, GF(qm) consists of elements: 0,1, α, α2,…,
αqm-2 [5].
Let us regard IRB with n=4, S=13 in terms of finite fild
theory. In this case prime element x of GF(32) satisfies
equation f(x)=x3-x-1, where f(x)- is 3-degree polynomial
irreducible over GF(32), p=3, m=2 (Fig. 1).
TABLE 1
INDEXES
n
4
8
16
32
64
128
256
512
OF THE OPTIMIZED IRB CORRECTING CODES
t≤
0
3
7
15
31
63
127
255
S
7
15
31
63
127
255
511
1025
Q
0
0,2
0,226
0,238
0,244
0,247
0,248
0,249
P
14
30
62
126
254
510
1022
2050
2
Õ +Õ
Table 1 shows both increase error correction capability of
the optimized cyclic IRB code in increasing order both of
Õ
code combination length and code size. That is, a maximal
Õ3
2
Õ +2Õ+1
number of corrected errors t theoretically possible reaches
Õ Õ1
quarter of code combination length, while maintaining on the
2
2Õ +2Õ+1
code size twice as large of the code combination length.
Õ
1
3
1
For example, the IRB with n=8, R=4 corresponding to the
2
table
given above consists of 1,1,1,2,2,1,3,4. The initial
2Õ +2
2
combination of the IRB correcting code follows from
Õ9
Õ +2
equation (1): 111010110010001. The complete set of code
Õ +2
2
combinations creates by cyclic shift of this combination with
2Õ +Õ+1
2
added number of cyclic shifts the same combination of
Õ +2Õ
converse symbols, namely, 000101001101110. So, we have
Fig.1. Graphic representation of IRB with n=4, S=13 in
got optimized IRB correcting code with t ≤ 3, S=15. and
terms of finite field by f(x)= x3- x- 1
P=30.
Õ+1
2
Õ +Õ+1
2
To see this we observe IRB with n=4, S=13 as a nonConclusion
uniform geometric figure of four (n=4) vertexes (1, x, x+1,
Application of the combinatorial configurations such as
x+2) which form ring sequence of positive integers (1, 2, 6,
Ideal Ring Bundles (IRB)s for innovative information
4) in field of S =13 vertexes.
Synthesis of the IRB correcting code starts from finding an technology provides techniques for improving quality
initial combination of the code using the equation (1) to indices of coding systems with respect to noise immunity
arrange four (n=4) symbols “1” of S =13 code positions, based on the combinatorial configurations theory. Research
into the underlying mathematical structures provides
while remaining ones – “0”:
development optimal error correction methodology for
j
synthesis high performance coding systems due to optimal
:
xj – 1 ≡
(1)
k i (mod S ), j=1,2,…, n,
placement of symbols “0” and “1” in code combinations of
i 1
the system using combinatorial techniques. The techniques
where kj is i –th element of the IRB.
The set of code combinations creates by cyclic shift of the for optimum synthesis of cyclic codes with improved
initial combination taking rest S - 1 combinations of possibilities for correcting errors based on cyclic groups in
extensions of Galois filds and theory of the IRBs. These
correcting code.
Minimum Hamming distance d for this correcting code is applications make it possible to configure information
systems for correcting a large number of errors, for example
defined as follows:
d = 2(n – R)
(2) communications and satellite.

If n = 2R than d = n, S = 2n-1, and number t of corrected
errors is function of n :
t ≤ ent (n/2 – 1)
(3)
Quality index Q of correcting capability of the optimized
IRB code defines error correction efficiency of the code. It is
relation maximal number of corrected errors t to code
combination length S :
Q= t/S
(4)
Code size P for this coding system is
P=2S
(5)
The outcome of the calculations is represented below.
References
[1] http://en.wikibooks.org.wiki/Data_Coding_Theory.
[2] http://en.wikibooks.org.wiki/BCH_Code
[3] V.V.Riznyk, Multidimensional Systems Based on
Perfect
Combinatorial
Models,
IEE,
Multidimensional Systems: Problems and Solutions,
London, No 225, 1998, pp.5/1-5/4.
“COMPUTER SCIENCE & INFORMATION TECHNOLOGIES” (CSIT’2014), 18-22 NOVEMBER 2014, LVIV, UKRAINE