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ПРИКЛАДНАЯ ДИСКРЕТНАЯ МАТЕМАТИКА
2023 Прикладная теория кодирования № 62
УДК 519.725 DOI 10.17223/20710410/62/8
WEIGHT DISTRIBUTION OF LOW-DENSITY PERIODIC RANDOM ERRORS AND THEIR CORRECTING CODES WITH ERROR DECODING PROBABILITY
L. Haokip, P. K. Das
Tezpur University, Tezpur, India
E-mail: h_letminthang@yahoo.com, pankaj4thapril@yahoo.co.in
We present weight distribution of low-density periodic random errors in the space of all q-ary n-tuples along with the average Hamming weight of the error set. We also provide necessary and sufficient conditions for the existence of linear codes correcting such error pattern. Examples of such codes are given. Finally, probability of decoding error of such codes over a binary symmetric channel is derived.
Keywords: parity-check matrix, syndromes, periodic random error, error decoding probability.
ВЕСОВОЕ РАСПРЕДЕЛЕНИЕ ПЕРИОДИЧЕСКИХ СЛУЧАЙНЫХ ОШИБОК МАЛОЙ ПЛОТНОСТИ И ИХ КОРРЕКТИРУЮЩИЕ КОДЫ С ВЕРОЯТНОСТЬЮ ОШИБКИ ДЕКОДИРОВАНИЯ
Л. Хаокип, П. К. Дас
Университет, Тезпур, г. Тезпур, Индия
Изучается весовое распределение периодических случайных ошибок малой плотности в пространстве всех q-арных n-кортежей и среднее число таких ошибок на блок длины п. Приведены необходимые и достаточные условия существования и примеры линейных кодов, исправляющих такие ошибки. Вычислена вероятность ошибки декодирования таких кодов для двоичного симметричного канала связи.
Ключевые слова: матрица проверки чёт,ноет,и, синдромы, периодическая случайная ошибка, вероятность ошибки декодирования.
1. Introduction
Periodic random error is one type of errors which occurs in electronic control unit like power lines, inverters, car electric, compact disc, CD ROM. This was observed by N. Lange in 1994 [1]. This error pattern behaves in such a way that any b consecutive components are disturbed after a gap of some fixed positions repeatedly. Linear codes capable of detecting and correcting such errors along with their Hamming weight distribution and decoding error probability are studied in [2, 3]. A periodic random error can be defined as follows.
Definition 1. An s-periodic random error of length b is an n-tuple whose nonzero
b
separated by s positions.
In 1963, Wvner [4] observed that for low intensity disturbances, only a few components within a burst [5] get disturbed, and he introduced the concept of low-density burst.
We extend this idea to periodic random errors whose intensity is low and define low-density periodic random error as below.
Definition 2. Low-density periodic random error is an s-periodic random error of length b such that each sets of b consecutive components separated by s zeros contains at most w, w ^ b, nonzero components.
Let £(s,b|w),n,q denote the set of all low-density periodic random errors in the space of n-tuples over GF(q). For example, the following vectors are some members of £(3,2|1),10,2:
0000001000,0000010000,0100000000,0100001000, 0100010000,
1000000000,1000001000,1000010000,0000000100, 0010000000, 0010000100,0010001000,0100000100,0000000010,0001000000, 0001000010, 0001000100,0010000010,0000000001,0000100000.
In this paper, we study the Hamming weight distribution of the vectors of £(s,b|w),n,q. Then we study the existence of linear codes correcting the errors from the set £(s,b|w),n,q. We denote a linear code that corrects low-density periodic random errors from the set £(s,b|w),n,q by LDP(s,b|w),n,qRC-eode. We further study the probability of decoding error for the errors set £(s,b|w),n,q over a binary symmetric channel. Throughout the paper, we consider n = A(b + s) + /, where 0 ^ l < s + b and A e N.
The organization of remaining part of the paper is as follows. Section 2 gives the Hamming weight distribution of the vectors of £(s,b|w),n,q along with examples. Average Hamming weight of the vectors of £(s,b|w),n,q is derived. In Section 3, we obtain necessary and sufficient conditions for existence of a LDP(s,b|w),n,qRC-code followed by three examples. Finally, we provide the probability of decoding error for the errors of £(s,b|w),n,q over a binary symmetric channel,
2. Hamming weight distribution of vectors of £(s,b|w),n,q In this section, we give the Hamming weight distribution of vectors of £(s,b|w),n,q and average Hamming weight of a vector from the set £(s,b|w),n,q.
Here n = A(b + s) + I, where 0 ^ l < s + b, then the maximum Hamming weight wmax of a vector of £(s,b|w),n,q is given by
wA, when l = 0,
Wmax = < wA + min{1, w}, when 1 ^ l < b,
w (A + 1), when b ^ l < s + b.
We first state the following lemma from [6].
Lemma 1 [6], If A, denotes the number of sets of non-zero positions and m, the maximum number of nonzero positions in a vector of £(s,b|w),n,q that starts from ith position, then
A,-
n — i + 1 s + b
and m,-
n — i + 1 s + b
b + y((n — i + 1) mod (b + s)),
where |_xj means the greatest integer ^ x, means the smallest integer ^ x, and
Y(r)
r, if 0 ^ r ^ b, b, if b < r < b + s.
Next, we give the following Lemma to derive weight distribution of vectors of £(s,b|w)
Lemma 2. Let p, be the number of common nonzero positions of the errors of £(s,b|w),n,q th ' starts from the 1st position. Then p, is given bv
that starts from the zth (i = s + 2,..., s + b) position with an error vector of £(s,b|w),n,q that
(1) when 1 = 0 and b — 1 ^ 1 ^ s -f b p, = (i — s — 1)$, and
(2) when 1 ^ 1 < b — 1: p, = n — i — b +1"
(i — s — 1)^-1 for i = s + 2, (i — s — 1)^j-1 + 1 for i = s + 3, s + 4,
,s + b,
where $
s+b
Proof.
Case 1 : 1
^d b - 1 ^ I ^ s + b.
The common nonzero positions of the error pattern of £(s,b|w),n,q that starts from the (s + 2)th position with the error pattern that starts from the P* position are s + b + 1,
"n_(s + 1)_b +1
2(s + b) + 1, ..., $s+i(s + b) + 1, where $s+i = ( + ) +
s+b
This number of
common nonzero position is given by ,5s+1.
The common nonzero positions of the error pattern of £(s,b|w),n,q that starts from the (s + 3)th position with the error pattern that starts from the 1st position are s + b + 1, s+b+2,
" n_(s + 3)_b +1
2(s+b) + 1, 2(s+b)+2, ..., &+2(s+b) + 1, &+2(s+b)+2, where &+3 = -(--
s+b
This number of common nonzero position is given by 2^s+2,
Continuing this, the common nonzero positions of the error pattern of £(s,b|w),n,q (s + b) 1
position are s + b +1 s + b + 2, ..., s + b + (b - 1) 2(s + b) + 1, 2(s + b) + 2,
2(s + b) + (b — 1), ..., $s+6-i(s + b) + 1 &+6-i(s + b) + 2 n — (s + b — 1) — b +1
1
(b — 1)&+6-i.
b
., ^s+b-i(s + b) + (b - 1), where This number of common nonzero position is given by
n — i — b +1 s + b
Thus p, = (i — s — 1)$i-i for i = s + 2, s + 3,..., s + b, where $
Case 2 : 1 ^ 1 < b — 1.
The common nonzero positions of the error pattern of £(s,b|w),n,q that starts from the
(s + 2)th position with the error pattern that starts from the P* position are s + b + 1,
n (s + 2) b + 1 2(s + b) + 1, ..., $s+i(s + b) + 1, where $s+i = ( + ) +
s+b
This number of
common nonzero position is given by $s+i.
(s + 3) 1
positions, are t + b +1 s + b + 2 2(s + b) + 1, 2(s + b) + 2, ..., $s+2(s + b) + 1, $s+2(s + b) + 2, "n — (s + 2) — b + 1"
where $
s+2
s+b
The common positions with the last set are As+3(s + b) + 1, As+3(s + b) + 2, ,,,, As+3(s + b) +1 (Aj are given by Lemma 1), whose number is I. Therefore, the total number of common nonzero positions is given by 2$s+2 + I.
(s + b) 1
set of nonzero positions, are s + b + 1, s + b + 2, ,,,, s + b + (b — 1), 2(s + b) + 1, 2(s + b) + 2,
s
2(s + b) + (b - 1), ..., &+b-i(s + b) + 1 As+b-i(s + b) + 2, ..., &+b-i(s + b) + (b - 1), n - (s + b - 1) - b + 1"
where Дз+ь-i
s + b
The last set has common positions As+b(s + b) + 1, As+b(s + b) + 2, ..,, As+b(s + b) + l, whose numte is l. This number of common nonzero position is given bv (b — 1)es+b-1 + l-(i — s — 1)вг-1 for i = s + 2, (i — s — 1)вг-1 + l for i = s + 3, s + 4,..., s + b. Lemma 2 is proven, ■
Thus p
Theorem 1. Let Rb^ (j) the total number of vectors of £(s,b|w),n,q, whose Hamming weight is j, Then: For j = 1:
s+i
R"b|w(1) = S
i=i
For 2 ^ j ^ w:
m.
ki-i 1
s+b
(q - 1)+ E
i=s+2
mi
ki-i 1
s+i
R"b|w(j) = E
i=i
mi j
ki-i j
s+b
(q - 1)j + E
i=s+2
mi j
ki-i j
ßi-i 1
PA + 1 Pi-i
(q - 1).
(q - 1)J
For W + 1 ^ j ^ Wmax - 1
s+i
R"b|w (j ) = E
i=i
mi j
s+b
+ E
i=s+2
mi j
ki-i j
Pi
ki-i j
+
(b - wO^i-A / mi - b
1 Д j - w - 1
(q - 1)j +
Pi-i
(b - w)ßi-i 1
mi - b j-w-1
(q - 1)J
For j = Wmax:
(s + 1)Ьл + Ьл-1 - s - 1 (q - 1)
R?b|w(Wmax) = < bA(q - 1)w
when l = 0, when 1 < l < b,
(l - b + 1)bA+1 + bA - l + b - 1 (q - 1)wmax, when b ^ l < s + b,
where ps+1 = 1 k0 = 0 k, = mj+1 - ,,, ,,, m^d p, are given by Lemmas 1, 2, Proof.
j=1
The number of error patterns of weight 1 that start from the ith positions, where i =
/ ^^^ - \ f m^fy - I \
= 1, 2,..., s + 1, is given by f j (q - 1) But in the calculation f j+j (q - 1), number of
m-
already counted nonzero components in i j (q-1) is k, = mj+1 -,, for i = 1, 2, 3,..., s+1,
where в =
n - i - b + 1 s + b
b
positions in which the nonzero elements of the error pattern start from the imposition
s+1
Therefore, the total number of the errors having weight 1 is
i=1
with k0 = 0,
mi 1
ki-i 1
(q - 1)i
1
1
j
j
w
max
max
For error patterns whose starting position is i = s + 2,..., s + b, all the weight 1 vectors
b
except the last set which may be less than b components). The number of these nonzero components is given by Thus, L -1 ), number of weight 1, need to be subtracted from
1
e a e a e a
'mA Ai-A 1 - 1- - 1-
given by the quantity
, We have
s+i
R™b|w (1) = S
j=i
m, 1
kj-i 1
s+b
(q — 1)+ E
j=s+2
mj 1
kj-i 1
$j-i 1
(q — 1).
Case 2: 2 ^ j ^ w.
As above, the total number of the errors having weight j that start from the ith positions,
'mA /kj-iN
s+i
where i = 1, 2,..., s + 1, is the quantity
j=i
j
j
(q — 1)j with ko = 0.
But, for error patterns that start from positions i = s + 2,..., s + b, there are some more
1
By Lemma 2, p denotes the number of common nonzero components that start from i (i = s + 2, .e. . , as + b)
the Brst position, j (q - tf gives the -umber of vectors of weight j rn the ^ (i = . + 2,
. . . , s + b) e a
position. This includes some vectors which are deleted by the term (q — 1)j, thus the
term j (9 — l)j is added to include such already deleted error etars. ..ere p,« = 1 So
the exact number of common vectors that need to be excluded is
Therefore, we have
R
n
s,b|w
s+1
(j)=E
j=1
mj
j
kj1
j
s+b
(q—1)j + E
j=s+2
mj j
kj-1 j
+
j
pj-1 j
(q — 1)J
(q—1)j
Case 3: w + 1 ^ j ^ wmax — 1,
In this case, we can also similarly calculate the total number of all error vectors having weight j and starting from the ith position, where i = 1, 2,..., s + 1, after deleting the common vectors as the quantity
s+1
E
j=i
mj j
kj-i j
(b — w)$j-i 1
mj — b j—w—1
(q — 1)j with ko = 0.
Again, for error vectors having weight j starting from (s + 2)th to (s + b)th positions,
av w v wt
pj
Pj-i
(q — 1)j, Therefore,
s+i
Rn,b|w (j ) = E
j=i
mj
s+b
+ E
j=s+2
mj j
kj-i
kj-i j
(b — w)$j-i 1
(b — w)$j_A / mj — b 1 ) V j v w — 1 mj — b j—w—1
(q — 1)j +
+
Pj-i
(q — 1)J
j
j
Case 4 : j = wmax-
j
formulas are found for / = 0, 1 ^ l < b, and b ^ l < s + b:
[(s + 1)bA + bA-1 — s — 1] (q — 1)w
;(Wmax) = < bA(q — 1)W
when I = 0, when 1 < I < b,
[(/ — b +1)bA+1 + bA — / + b — 1] (q — 1)wmax, when b ^ / < s + b.
Theorem 1 is proven, ■
Remark 1. The values of m, and p, in [3] are given bv
bA
for 1 ^ i ^ s + 1,
m,-
bA + s — i + 1 for s + 2 ^ i ^ s + b,
m,
bA + I — i + 1 bA
for 1 ^ i ^ /,
for I + 1 ^ i ^ s + I + 1,
bA + s + / — i + 1 for s + / + 2 ^ i ^ s + b,
b(A +1)
for 1 ^ i ^ I- b + 1,
m,
b(A + 1) + (/ — b — i + 1) for I — b +1 < i ^ /,
if / = 0,
if 1 < / < b,
if b ^ / < s + b,
bA
for 1 + I ^ i ^ s + b, if I = 0 or b ^ / < s + b,
P,
i(p1 — 1) + /for / + 1 ^ i ^ b — 1,
for 1 ^ i ^ /,
if 1 < / < b.
In this paper, we consider the simplified form for m, and p, in Lemma 1 and Theorem 1, b=w
bA, when / = 0,
< bA + /, when 1 ^ / < b, b (A + 1), when b ^ / < s + b,
and
(b — w)ßi-i 1
m, — b j — w — 1
Then Theorem 1 coincides with Lemma 3,1 [3].
Example 1. Considering q = 3 n =11 s we have A = 2, l = 11 mod 5 = 1 m1 = 5 m2 ,o = ,1 = ■ ■ ■ = ,4 = 2. Then
3 b = 2, and w = 1 in Theorem 1, ■ ■ ■ = m5 = 4 Ps+1 = 1 Ps+2 = 2,
Rl12|1(1)
+
(3 — 1)1 +
0
(3 — 1)1 +
(3 — 1)1 +
R112|2(2)
+
3
2
(3 —1)1+
(3 — 1)1
5 ■ 2 + 2 ■ 3 ■ 2 + 0 = 22;
2) — (<3r%_ ! — !
(3 — 2) ■ 2)( 2 1
(3 — 1)2+ (3 — 1)2+
max
max
0
+
+
+
(3 — 2)■ 2 (3 — 2)■ 2
211
1
211
(3 — 1)2+
(3 — 1)2+
(3 — 2)■ 2 1
211
(3 — 1)2
76;
R1,2|3(3) = 22(3 — 1)3 = 32.
Here the maximum weight is wm in the next section.
3
Example 2. Considering q = 2, n =12 s = 3 b have A = 2, l = 12 mod 5 = 2, m1 = 6, m2 = 5 m3 = Ao = A = ■ ■ ■ = A = 2. Then
R^n (1)
^^d w = 1 in Theorem 1, we
■ = m,5 = 4, Ps+1 = 1 Ps+2 = 2,
+
+
+
+
+
(3 — 2) ■ 2 1
+
2 - 1- 1
+
12 + 5 + 3 ■ 2 + 2 = 25;
R112|3(3) = (2 — 2 + 1) 23 + 22 — 2 + 2 — 1 = 11.
wmax = 3
Example 3. Taking q = 2 n =14 s = 4 b = ^^d w = 2 in Theorem 1, we have A = 2 l = 14 mod 7 = 0, m1 = m2 = ■ ■ ■ = m5 = 6, mg = 5 m7 = 4 ps+1 = 1 P,+2 = 1
Ps+3 = 2 & = A = ■ ■ ■ = A = 2 & = A = 1. Then
R4,3| 1(1)
0
+
+ 6
4
+
+
+
+
+
5 2
R143 12 (2)
Y6a e4
a 2 2 1a 1
2+2
+
4 4
6a—(2
6
+ 6
4
+
+
+
+
+ 4a
6
+
21 2+2
6 + 4 ■ 2 + 0= 14;
+
4a 2
+
= 15 + 9 ■ 4 + 4 + 2 = 57;
2
2
2
+
R143|3(3) =
(3 — 2) ■ 2 1
(3 — 2)■ 2
1
4
3-2-1
+
4
+
+
+
3—2—1 4 3
+
+
+
+
) —(4) — (P—10'2
(3 — 2) ■ 2)( 4 1 J V3 — 2 - 1
1) T —f — 2 —:
3) — ((3 -1 -:
4
3 - 2- 1
+
+
= (20 - 2) + (20 - 4 - 2) ■ 4 + (10 - 4 - 1) + (4 - 1 - 1) = 81; R143|4(4) = (4 + 1) ■ 32 + 32-1 - 4 - 1 = 43.
wmax = 4
Theorem 2. The average weight of a vector of the set £(s,b|w),n,q is
Wmax / w_max
E (j) E (j),
j=1
j=1
where Rnb|w (j) '1S given by Theorem 1,
Proof. By Theorem 1, the number of vectors of £(s,b|w),n,q having Hamming weight j is
wmax
Rnb|w(j) and the total weight of all vectors of £(s,b|w),n,q is given by Y1 jRnb|w(j)■ The ratio
j=1
gives the required average weight, ■
3. Existence of LDP(s,b|w),ra,qRC-codes
q
LDP(s,b|w),ra,qRC-codes. We also derive an upper bound on the number of codewords for such a code. We also construct examples based on the results. Theorem 3. Every (n, k) LDP(s,b|w),n,qRC-code satisfies
n — k ^ logq
1 + E R.biw (j)
j=1
where (j) '1S given by Theo rem 1.
Proof. By Theorem 1, the number of error vectors of £(s,b|w),n,q including the zero
Wmax
vector is 1 + |£(s,b|w),n,q| = 1 + E the maximum available coset is qn-k and
j=1 , |
LDP(s,b|w),n,qRC-code corrects all such errors, we have
qn-k ^ 1+ E R?Ww(j)
n — k ^ logq
j=1
1+ E Rn.biw (j)
j=1
Theorem 3 is proven, ■
Remark 2. The maximum number of codewords of an (n, k) LDP(s,b|w),n,qRC-code is
qn
M C
1 + E Rn.biw (j)
j=1
In the following theorem, we apply the well known technique used in Varshamov — Gilbert — Sacks bound (see [7] and [8, Theorem 4,7]),
Theorem 4. For existence of an (n, k) LDP(s,6|w) ,n,qRC-code, the following condition is sufficient:
w 1 .b- 1a / w eb- 1a AA 1min{w,g} e^a / wmax
q> S (—)(q—1)ls (—)(q—1j S 0<9—1)j (1+ S R"-w(j'
ea
where g = 7(l) and R^—l(j) '1S giyen by Theorem 1, Here ( • ) (q — 1)j = 1 for g = 0,
, | " j=o \jJ
Proof. The proof is done by constructing an appropriate (n — k) x n parity-check matrix H of the code. Suppose that the first n — 1 columns h1, h2, h3,..., hn-1 are added suitably to H, Then any (nonzero) column is added to H provided that it is not a linear combination of at most w — 1 columns among the immediately preceding b — 1 columns
wb sb wb gap of s columns confined to the first n — b columns (the last set may contain less than b columns). This can be written as
/6-1 6—1 6—1 g—1 \
= I E ai1hn—i + E bi 1 hn— (s+6)—i + E bi2hn—2(s+6)—i + ... + E biAhn—A(s+6) —i I + \i=1 i=0 i=0 i=0 /
(6—1 6—1 6—1 g' —1 \ ^
ai1hj'—i + Ai1hj' — (s+6)—i + E Ai2hj' — 2(s+6)— i + ... + E Aa''—A'(s+6)—i i=0 i=0 i=0 i=0
where a,, bij-, a,, A, S GF(q) such that the number of nonzero a, is at most w — 1, and that of b,, a,, A, '1S at most w j' ^ n — b; g = 7(n mod (s + b)) = y(1), g' =
= 70(n — b — j' + 1) mod (s + b)t, and A' = n—b
e sa+ b
w—1/ b — 1\
The number of coefficients ai1 is ^ (q — 1)j.
¿=0 V j /
/ w /b — n A min{w,g} /A The number of coefficients b, is YM (q — 1)j Y] (q — 1)j, So the
Vj=A j / / ¿=0 W
number of all possible linear combinations in the first bracket of the right-hand side (2) is
w—1 eb- 1a ( w .b — 1a \A 1min(w,g} .A
,101 j A—jM j A—, 5 j(q—1)
The second bracket in (2) gives the total number of low-density periodic random error
wmax
in a vector of length n — b. This is given by Theorem 3 as 1 + Rn—iw(j)- Therefore, the
¿=1 s |w
total number of all the possible linear combinations of the right-hand side (2) is
A-1
5 A ^Usr—A-A 5' j1)j AS°>i- (3)
w— 1 f b— 1\ . ( w /b—1\ \ minj{w,g} f g\ I wmax
llq—lr 1 '+ > Rs,6|w
Since we have at most qn k columns, so taking qn k greater than or equal to the term computed in (3) gives the sufficient condition for the existence of the required code, ■
In the following examples, À', p-, and fij represent the values of A, p^ and fij respectively, when n is replaced by n — b.
Example 4. Consider n =11 s = 3, b = 2, w = 1, and q = 3 in Theorem 4, then
A = 2 l = 11 mod 5 = 1 A' = 1 pS+1 = 1 pS+2 = 1 fi0 = fi1 = fi2 = 2 fi3 = fi4 = 1-
Putting these values in the inequality (1), we get
0 /1\ ( 1 /2-1\ \ 1 min{w=1,g=1} /i\ / 3 \
3>SO(3—1)Hs(— )(3—1j s j(3—1)jl1+s=
= 1 ■ 3 ■ 3 (1 + 66) [Using Theorem 1 and Example l] = 603.
This implies n — k ^ 6. Thus, we can construct a parity eheck matrix H of order 6 x 11, which generates the (11,5) ternary LDP(3,2|1),11,3RC-code:
1 0 2 1 0 1 0 0 1 0 0
2 1 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 2 0 1 0 1
0 1 0 1 2 0 0 1 1 0 0
0 0 1 1 0 0 0 0 1 1 0
1 0 0 1 0 0 0 0 1 0 2
It can be verified from the Error Pattern-Syndromes Table 1 that the syndromes of all the errors are nonzero and distinct, showing that the code is a (11, 5) ternary LDP(3,2|i),n,3RC-code.
Table 1
Error Pattern-Syndrome
Error Patterns Syndromes Error Patterns Syndromes
00 000 00 000 1 001002 20 000 02 000 2 210000
00 000 00 000 2 002001 10 000 10 000 1 221000
00 000 01 000 0 002000 10 000 10 000 2 222000
00 000 02 000 0 001000 10 000 20 000 1 021000
00 000 10 000 0 100000 10 000 20 000 2 022002
00 000 20 000 0 200000 20 000 10 000 1 011001
00 000 01 000 1 000002 20 000 10 000 2 012000
00 000 01 000 2 001001 20 000 20 000 1 111001
00 000 02 000 1 002002 20 000 20 000 2 112000
00 000 02 000 2 000001 0 00 000 01 000 000100
00 000 10 000 1 101002 0 00 000 02 000 000200
00 000 10 000 2 102001 0 01 000 00 000 200010
00 000 20 000 1 201002 0 02 000 00 000 100020
00 000 20 000 2 202001 0 01 000 01 000 200110
01 000 00 000 0 010100 0 01 000 02 000 200210
02 000 00 000 0 020200 0 02 000 10 000 100120
01 000 00 000 1 011102 0 02 000 20 000 100220
01 000 00 000 2 012101 0 01 000 10 000 202010
02 000 00 000 1 021202 0 01 000 20 000 201010
02 000 00 000 2 022201 0 02 000 10 000 102010
01 000 01 000 0 012100 0 02 000 20 000 101020
01 000 02 000 0 011100 0 10 000 01 000 010200
02 000 01 000 0 022200 0 10 000 02 000 010000
End of Table 1
Error Patterns Syndroms Error Patterns Syndromes
02 000 02 000 0 021200 0 20 000 10 000 020000
01 000 10 000 0 110100 0 20 000 20 000 020100
01 000 20 000 0 210100 00 00 000 01 00 111111
02 000 10 000 0 120200 00 00 000 02 00 222222
02 000 20 000 0 220200 00 01 000 00 00 100111
01 000 01 000 1 010102 00 02 000 00 00 200222
01 000 01 000 2 011101 00 01 000 01 00 211222
01 000 02 000 1 012102 00 01 000 02 00 022000
01 000 02 000 2 010101 00 02 000 01 00 011000
02 000 01 000 1 020202 00 02 000 02 00 122111
02 000 01 000 2 021201 00 01 000 10 00 100211
02 000 02 000 1 022202 00 01 000 20 00 100011
02 000 02 000 2 020201 00 02 000 10 00 200022
01 000 10 000 1 111102 00 02 000 20 00 200122
01 000 10 000 2 112101 00 10 000 01 00 011121
01 000 20 000 1 211102 00 10 000 02 00 122202
01 000 20 000 2 212101 00 20 000 01 00 211101
02 000 10 000 1 121202 00 20 000 02 00 022212
02 000 10 000 2 122201 000 00 000 01 0 000010
02 000 20 000 1 221202 000 00 000 02 0 000020
02 000 20 000 2 222201 000 01 000 00 0 000200
10 000 00 000 0 120001 000 02 000 00 0 000100
20 000 00 000 0 210002 000 01 000 01 0 000210
10 000 00 000 1 121000 000 01 000 02 0 000220
10 000 00 000 2 122002 000 02 000 01 0 000110
20 000 00 000 1 211001 000 02 000 02 0 000120
20 000 00 000 2 212000 000 01 000 10 0 112011
10 000 01 000 0 122001 000 01 000 20 0 222122
10 000 02 000 0 121001 000 02 000 10 0 111211
20 000 01 000 0 212002 000 02 000 20 0 222022
20 000 02 000 0 211002 000 10 000 01 0 100121
10 000 10 000 0 220001 000 10 000 02 0 100101
10 000 20 000 0 020001 000 20 000 01 0 200202
20 000 10 000 0 010002 000 20 000 02 0 200212
20 000 20 000 0 110002 0000 01 000 10 100010
10 000 01 000 1 120000 0000 01 000 20 100020
10 000 01 000 2 121002 0000 02 000 10 200010
10 000 02 000 1 122000 0000 02 000 20 200020
10 000 02 000 2 120002 0000 10 000 01 001202
20 000 01 000 1 210001 0000 10 000 02 002201
20 000 01 000 2 211000 0000 20 000 01 001102
20 000 02 000 1 212001 0000 20 000 02 002101
Example 5. Consider n =12 s = 3 b = 2 w = 1, and q = 2 in Theorem 4, then A = 2 l = 12 mod 5 = 2 A' = 2 p's+1 = 1 Ps+2 = 1 = A = ■ ■ ■ = A = 2 A = 1. From inequality (1), we get
0 /9_1\ / 1 /2 —1\ \ 1 min{w=1,g=2} /o\ /3 \
>g( j ^ - E Q1+ EW
J
1 ■ 21 ■ 3(1 + 24) = 150
H
This implies n — k ^ 8 which gives rise to a (12, 4) binary LDP^^^RC-code and its parity check matrix H is given by
" 1000000001 1 1 " 010000000101 001000001110 000100001101 000010000110 000001001100 000000100111 000000010101
It can be verified like the previous one that the syndromes of all the errors are nonzero and distinct, so the code is a (12,4) binary LDP(3,2|1),12,2RC-code.
Example 6. Consider n =14 s = 4 b = 3 w = 2, and q = 2 in Theorem 4, then A = 2 l = 14 mod 7 = 0 A' = 1 p^+i = 1 pS+2 = 1 pS+3 = 2 $ = = ^2 = 2,
P'3 = = = A6 = 1- Inequality (1) gives
2-1 /3_l\ / 2 \ 1 min{w=2,0=0} ,
2n-k> £( )(2-1)j( E (3-1)(2-1)0 £ I9 1(2-1)^ 1 + E «(j)
j=o V J
j=0
j=0
j=1
3 ■ 41 ■ 1 (1 + 135) = 1632.
This implies n — k ^ 11 which leads to a (14, 3) binary LDP(3,2|2),14,2RC-code with parity check matrix
H
10000000000111 01000000000011 00100000000010 00010000000010 00001000000011 00000100000110 00000010000110
00000001000011 00000000100110 00000000010011 00000000001011
Here we can also verify that all error patterns give nonzero and distinct syndromes.
Finally, we give the probability of decoding error for a LDP(s,b\w),n,qRC-code over a binary symmetric channel.
Theorem 5. Let PDR(E) be the probability of decoding error of an (n, k) binary LDP(s,b|w),n,2RC-code on a binary symmetric channel with transition probability e, then
PDr(E) = 1 — E Rnb\w(j) ej(1 — e)n j, where R" b\w(j) '1S given by Theorem 1. j=1
Proof. Since the binary symmetric channel has the transition probability e, the probability of occurring of any one of the error vector of weight j is ej (1 — e)n-j. So the
w_max
probability of occurring of any error vector from the set £(s,b\w),n,q is E R" b|w(j) ej(1 —e)n-j,
j=1 " 1
Since the code corrects all such error patterns, the probability of a decoding error of the
wmax
code is PDr(E) = 1 - m RV(j) ej(1 - e)n-j. ■
j=i 1
Remark 3. For s = 3 b = ^^d e = 0.1, we determine the probability of decoding error PDr(E) of binary LDP(S)b|w))ra;2RC-codes of different lengths as follows (Table 2),
Table 2 Values of PDr(E)
n A I PDr(E)
10 2 0 0.19
11 2 1 0.21
12 2 2 0.23
13 2 3 0.29
14 2 4 0.31
15 3 0 0.33
We find that the probability of decoding error of LDP(S)b|w))ra;qRC-code increases as the length of the code increases. So a smaller length code is more efficient,
4. Conclusion
This paper derives the weight distribution of low-density periodic random errors. Then necessary and sufficient conditions for the existence of linear codes that correct such errors, along with the error decoding probability of the codes, are presented. It can be interesting to explore some more systematic methods by which we can construct such codes. We can also investigate array code or cyclic code instead of linear code that can deal with such errors,
5. Acknowledgement
The first author is supported by JEF fellowship from Council of Scientific and Industrial Research, India (File No. 09/796(0085)/2018-EMR-I).
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