Wei_Zhou_Exit_Seminar_b

Maximum True Burst-Correcting Capability of Fire
Codes, Fire-BCH Codes, and BCH Codes for the
Rosenbloom-Tsfasman Metric
Wei Zhou
Department of Electrical and Computer Engineering
University of California, Davis
Davis, CA 95616, U.S.A.
May 25, 2016
Wei Zhou Exit Seminar 1 / 66

Research Topics
1 The true burst-correcting capability of Fire codes
2 Fire-BCH codes
3 BCH codes for the Rosenbloom-Tsfasman metric

The true burst-correcting capability of Fire codes
We classify Fire codes based on their parameters.
We derive necessary and suﬃcient conditions for a Fire code to
correct a burst of a given length.
We give lower and upper bounds for each class of Fire codes.
We construct speciﬁc codes that achieve those bounds.

Introduction
Communications channels may introduce errors in a localized inter-
val, we call such errors a burst because they occur in consecutive
bits.

Introduction
Communications channels may introduce errors in a localized inter-
val, we call such errors a burst because they occur in consecutive
bits.
A binary vector of length n is called a burst of length b if the 1’s are
conﬁned to b, but not less than b, consecutive positions.

Burst correcting codes
A code is said to have burst correcting capability of b if it can
correct any error vector which is a burst up to length b.
The burst correcting capability b of an [n, k] linear code must satisfy
the following inequalities:
2b ≤ r Reiger bound
n ≤ 2r−b+1
− 1 Abramson bound
where r = n − k.
For cyclic codes, a simple decoding algorithm, known as burst trap-
ping, can decode all bursts up to their burst correcting capability.

Fire codes
A Fire code is a cyclic code generated by
g(x) = (x2b∗−1 + 1)g0(x) over F2, where
g0(x) is an irreducible polynomial, diﬀerent from x, of degree
m ≥ b∗
, and period n0, n0 2b∗
− 1.
The Fire code has a length of LCM(2b∗
− 1, n0).
The designed burst correcting capability of the code is b∗
.
How about its true burst correcting capability?

An interesting observation
b∗ = 2 and g0(x) is an irreducible polynomial of degree 10 and period
1023.
Code length: n = LCM(2b∗
− 1, n0) = 1023.
Redundancy: m + 2b∗
− 1 = 13.
Designed burst correcting capability b∗
= 2.
However, diﬀerent irreducible polynomials may give diﬀerent true
burst correcting capability b.
g0(x) b
x10 + x3 + 1 2
x10 + x8 + x3 + x2 + 1 3
x10 + x5 + x3 + x2 + 1 4

Problem formulation
Recall n0 is the period of g0(x).
Let h0 = GCD(2b∗
− 1, n0),
Let t = 2m
−1
n0
,
where m is the degree of g0(x) and n0 is its period.
Fire codes with parameters b∗, h0, and t are called (b∗, h0, t)-Fire
code.

Problem formulation
Recall n0 is the period of g0(x).
Let h0 = GCD(2b∗
− 1, n0),
Let t = 2m
−1
n0
,
where m is the degree of g0(x) and n0 is its period.
Fire codes with parameters b∗, h0, and t are called (b∗, h0, t)-Fire
code.
Given b∗, h0 and t, what is maximum possible true burst correcting
capability, denoted by bmax(b∗, h0, t), of a (b∗, h0, t)-Fire code?

Theorem
The class of (b∗, h0, t)-Fire codes is nonempty if and only if
b∗, h0, and t are positive integers, h0 and t are odd, h0 divides
2b∗ − 1, and

Theorem
The class of (b∗, h0, t)-Fire codes is nonempty if and only if
b∗, h0, and t are positive integers, h0 and t are odd, h0 divides
2b∗ − 1, and
ordh0t(2) > ordh0t(2) for any divisor h0 = h0 of 2b∗ − 1 which is
divisible by h0.
If these conditions hold, then there is an inﬁnite number of (b∗, h0, t)-Fire
codes.
Note: orda(2), for an odd positive integer a, is the least positive integer
i such that 2i − 1 is divisible by a.

For example, for b∗ = 8, there are four divisors of
2b∗ − 1 = 15 : 1, 3, 5, and 15.
There is an inﬁnite number of (8, 15, 1)-Fire codes since there does
not exist a divisor h0 = h0 of 2b∗
− 1 which is divisible by h0 = 15.
There is no (8, 5, 1)-Fire code since 15 is divisible by h0 = 5 and
ord15(2) = ord5(2) as both equal 4.

For example, for b∗ = 8, there are four divisors of
2b∗ − 1 = 15 : 1, 3, 5, and 15.
There is an infinite number of (8, 15, 1)-Fire codes since there does
not exist a divisor h0 = h0 of 2b∗
− 1 which is divisible by h0 = 15.
There is no (8, 5, 1)-Fire code since 15 is divisible by h0 = 5 and
ord15(2) = ord5(2) as both equal 4.
There is an infinite number of (8, 3, 1)-Fire codes since ord15(2) >
ord3(2) = 2 and 15 is the only divisor of 2b∗
−1 = 15 that is divisible
by h0 = 3 and not equal to h0.
There is an infinite number of (8, 1, 1)-Fire codes since ordh (2) >
ord1(2) = 1 for every divisor h of 15 other than 1.

Theorem
Let bmax(b∗, h0, t) be the maximum true burst-correcting capability of
(b∗, h0, t)-Fire codes.
bmax(b∗
, h0, t) ≤
2b∗ − 1 − d(b∗, h0), if h0 < 2b∗ − 1,
2b∗ + log2 t , if h0 = 2b∗ − 1.
where d(b∗, h0) is the largest proper divisor of 2b∗ − 1 that is divisible
by h0.

Theorem
For primitive g0(x), t = 1, and
bmax(b∗
, h0, 1) ≤



b∗, if h0 = 1,
2b∗ − 2, if 1 < h0 < 2b∗ − 1,
2b∗, if h0 = 2b∗ − 1, (Abramson bound)

Some notations
Let v = (v0, v1, . . . , vn−1) be a vector of length n. It can be rep-
resented by the polynomial, v(x) = v0 + v1x + . . . + vn−1xn−1.
We use [f(x)]g(x), [f]g to denote the remainder polynomial obtained
by dividing polynomial f(x) by g(x) = 0, and the remainder obtained
by dividing integer f by g = 0, respectively.
[f(x)]g(x) ≡ f(x) mod g(x), [f]g ≡ f mod g
Let Bb be the set of nonzero polynomials over F2 of degree less than
b with constant 1, i.e.,
Bb = {B(x) ∈ F2[x] : degB(x) < b, B(0) = 1}

b-burst correcting cyclic code
A cyclic code generated by g(x) of length n is b-burst correcting iﬀ
it has no nonzero code polynomial of the form
[B1(x) + xl
B2(x)]xn−1,
where B1(x) and B2(x) ∈ Bb.
That means for any polynomial of the form
[B1(x) + xlB2(x)]xn−1, we have
g(x) [B1(x) + xl
B2(x)]xn−1.

Theorem
Consider the nontrivial Fire code generated by (x2b∗−1 + 1)g0(x), where
g0(x) is an irreducible polynomial of degree m ≥ b and period n0. The
code is b-burst-correcting if and only if the following is true: If

Theorem
1 B1(x) + xl B2(x) is divisible by x2b∗−1 + 1, where 0 ≤ l < 2b∗ − 1
and B1(x), B2(x) ∈ Bb are distinct polynomials unless l is a
nonzero multiple of h0, and

Theorem
2 B1(x) + xl0 B2(x) is divisible by g0(x), where 0 ≤ l0 < n0, then
l0 ≡ l (mod h0).

Example
Let a Fire code generated by (x2b∗−1 + 1)g0(x) where g0(x) is an
irreducible polynomial of degree m ≥ b∗. Then, the code is a
b∗-burst correcting code. For the code to be (b∗ + 1)-burst
correcting, two requirements are needed.
1 m ≥ b∗ + 1
2 If any of the following incongruences
(1 + xb∗
) + xl0 (1 + xb∗−1
) ≡ 0 (mod g0(x))
(1 + x + xb∗
) + xl0 (1 + xb∗−1
+ xb∗
) ≡ 0 (mod g0(x))
holds for some l0, 0 ≤ l0 < n0, then l0 ≡ b∗ (mod h0).

If g0(x) is an irreducible polynomial of degree m and period n0,
then
g0(x) =
m−1
i=0
(x − α2it
),
for some primitive element α ∈ F2m .
Let
p(x) =
m−1
i=0
(x − α2i
).
We say that p(x) is a primitive polynomial associated with g0(x).

Lemma
Let f(x) be a nonzero polynomial over F2 and g0(x) is an irreducible
polynomial of degree m and period n0 = (2m − 1)/t. Then, f(x) is
divisible by g0(x), iﬀ f(xt) is divisible by an associated primitive
polynomial p(x).

Lemma
Let f(x) be a nonzero polynomial over F2 and g0(x) is an irreducible
polynomial of degree m and period n0 = (2m − 1)/t. Then, f(x) is
divisible by g0(x), iﬀ f(xt) is divisible by an associated primitive
polynomial p(x).
Let l = l0t, then, the following two statements are equivalent:
B1(x) + xl0 B2(x) ≡ 0 (mod g0(x))
B1(xt
) + xl
B2(xt
) ≡ 0 (mod p(x))
Furthermore, l0 ≡ l (mod h0) ⇔ l ≡ tl (mod h) as h = th0.

Theorem
g0(x) is an irreducible polynomial of degree m ≥ b and period n0 =
(2m − 1)/t with associated primitive polynomial p(x). The code is b-
burst-correcting if and only if the following is true: If

Theorem
burst-correcting if and only if the following is true: If
2 B1(xt) + xl B2(xt) is divisible by p(x), where 0 ≤ l < 2m − 1,
then
l ≡ tl (mod h).

The AES condition
For each polynomial B(x) ∈ Bb, we deﬁne its index, ap(x)(B(x))
with respect to the primitive polynomial p(x) by
B(x) ≡ xap(x)(B(x))
(mod p(x)), 0 ≤ ap(x)(B(x)) < 2m
− 1
For example, ap(x)(1 + x + x3) = 7, where p(x) = 1 + x + x4.

The AES condition
For each polynomial B(x) ∈ Bb, we deﬁne its index, ap(x)(B(x))
with respect to the primitive polynomial p(x) by
B(x) ≡ xap(x)(B(x))
(mod p(x)), 0 ≤ ap(x)(B(x)) < 2m
− 1
For example, ap(x)(1 + x + x3) = 7, where p(x) = 1 + x + x4.
Then, the following two congruences are equivalent:
B1(xt
) + xl
B2(xt
) ≡ 0 (mod p(x))
xap(x)(B1(xt))
+ xl +ap(x)(B2(xt))
≡ 0 (mod p(x)).
for 0 ≤ l < 2m − 1, which is equivalent to
ap(x)(B1(xt
)) − ap(x)(B2(xt
)) ≡ l (mod 2m
− 1).
We call the integer incongruences corresponding to polynomial
incongruences the Abramson-Elspas-Short (AES) conditions
associated with given b∗, h0, t, and b > b∗.

Theorem
burst-correcting if and only if the following is true:
If B1(x) + xl B2(x) is divisible by x2b∗−1 + 1, where
0 ≤ l < 2b∗ − 1 and B1(x), B2(x) ∈ Bb are distinct polynomials
unless l is a nonzero multiple of h0,
then
ap(x)(B1(xt
)) − ap(x)(B2(xt
)) ≡ tl (mod h).

We can view the indices of the irreducible polynomials arising from
the incongruences
ap(x)(B1(xt
)) − ap(x)(B2(xt
)) ≡ l t (mod h)
as unknowns.
The incongruences depend on b∗, h0, t and b.
If these incongruences do not have a solution, then there is no
(b∗, h0, t)-Fire code which is b-burst-correcting.
If these incongruences have a solution, then
there is a “good” primitive polynomial, p(x), i.e., the indices with
respect to which satisfy the incongruences.
there is a (b∗
, h0, t)-Fire code which is b-burst-correcting

If m is suﬃciently large, then there are many primitive polynomials,
p(x), of degree m.
At least one of these primitive polynomials gives indices that satisfy
these incongruences.
By increasing m, there is an inﬁnite number of (b∗, h0, t)-Fire codes
which are b-burst-correcting.
Conclusion: bmax(b∗, h0, t) is the maximum value of b for which the
incongruences have a solution.

Example
We derive the AES conditions for previous example for the Fire code
to be (b∗ + 1 = 6)-burst correcting in case of b∗ = 5, h0 = 3, and
t = 1.
ap(x)(1 + x5
) − ap(x)(1 + x4
) ≡ 5 (mod 3)
ap(x)(1 + x + x5
) − ap(x)(1 + x4
+ x5
) ≡ 5 (mod 3)
By factoring the polynomials into a product of irreducible
polynomials and using the fact that ap(x)(B(x)) ≡ i ap(x)(fi(x))
(mod h), where B(x) = Π
i
fi(x), we obtain the AES conditions as
follows:
ap(x)(1 + x + x2
+ x3
+ x4
) ≡ 2 (mod 3)
ap(x)(1 + x2
+ x3
) − ap(x)(1 + x + x3
) ≡ 2 (mod 3)

Example - cont.
These two incongruences have precisely 12 solutions modulo three
for ap(x)(1+x+x2 +x3 +x4), ap(x)(1+x2 +x3), ap(x)(1+x+x3).
Next, we look for an irreducible polynomial p(x) of degree m such
that h0 = GCD(2b∗ − 1, 2m − 1) = GCD(9, 2m − 1) = 3, that meets
any one of these solutions.
The irreducible polynomial that meets these solutions gives burst
correcting capability of 6.
g0(x) b
1 + x2 + x3 + x4 + x8 5
1 + x3 + x5 + x6 + x8 6

Theorem
• bmax(b∗
, h0, t) ≥ b∗
+ (h0 − 1)/2.
• If b∗
≡ 2 (mod 3), then bmax(b∗
, (2b∗
− 1)/3, t) =
2(2b∗ − 1)
3
.
• bmax(b∗
, 2b∗
− 1, t) ≥ 2b∗
with equality if t = 1.

bmax(b∗
, h0, 1) for 5 ≤ b∗
≤ 8 and all possible values of h0
b∗ bmax(b∗, h0, 1)
5
10 if h0 = 9, i.e., m ≡ 0 (mod 6)
6 if h0 = 3, i.e., m ≡ 2 or 4 (mod 6)
5 if h0 = 1, i.e., m ≡ 1, 3, or 5 (mod 6)
6
12 if h0 = 11, i.e., m ≡ 0 (mod 10)
6 if h0 = 1, i.e., m ≡ 0 (mod 10)
7
14 if h0 = 13, i.e., m ≡ 0 (mod 12)
7 if h0 = 1, i.e., m ≡ 0 (mod 12)
8
16 if h0 = 15, i.e., m ≡ 0 (mod 4)
11 if h0 = 3, i.e., m ≡ 2 (mod 4)
8 if h0 = 1, i.e., m ≡ 1 or 3 (mod 4)

Table: Fire codes attaining bmax(b∗
, h0, t) = 2b∗
− 1 − d(b∗
, h0) with h0 = 1.
b∗ h0 t bmax(b∗, h0, t) d(b∗, h0) g0(x)1
3 1 3 4 1 127
4 1 11 6 1 2413
5 1 7 6 3 1231
6 1 217 10 1 152433
7 1 1197 12 1 1052465
8 1 23 10 5 4757
1
For each code, we write g0(x) in the octal notation. For example,
g0(x) = x6
+ x4
+ x2
+ x + 1 is written as 127.

Table: Fire codes with burst-correcting capability, b, exceeding twice their
designed burst-correcting capabilities.
b∗ h0 t g0(x) bmax(b∗, h0, t)
3 5 91 10011 ≥ 7
3 5 465 7647505 ≥ 8
3 5 1705 6406005 ≥ 9
3 5 2255 5017111 ≥ 10
3 5 6355 6130725 ≥ 11
3 5 416179 2755450655 ≥ 12
3 5 1248537 3657555473 ≥ 13

Research Topics
Fire-BCH codes
BCH Codes for the Rosenbloom-Tsfasman metric

Fire-BCH codes
Develop codes that can work over the compound burst/random error
channel.
A Fire-BCH code is a cyclic code generated by
g(x) = (x2b∗−1 + 1)g0(x) over F2, where
g0(x) co-prime with x2b∗
−1
+ 1.
g0(x) =
v∗
i=1 g1(x)g2(x) . . . gv∗ (x), gi(x) is a distinct irreducible
polynomial of degree m, and period ni for 1 ≤ i ≤ v∗
.
The Fire-BCH code has a length of LCM(2b∗
− 1, n0).
If g0(x) has α, α2, . . . , α2v∗
as 2v∗ consecutive roots, then the code
is guaranteed to correct v∗ random errors in any n0 consecutive
positions.

Research Topics
Fire-BCH codes

Reliable-to-unreliable channels and matrix codes
Reed-Solomon and BCH matrix codes
Properties
Cyclic structure of matrix Reed-Solomon codes
Dual codes of matrix Reed-Solomon and BCH codes

Reliable-to-unreliable channels
Consider a scenario where we have n parallel q-ary reliable-to-unreliable
channels, W0, W1, . . . , Wn−1. Transmit m symbols on each channel,
totally N = mn q-ary symbols are transmitted.

Matrix codes
A q-ary matrix code is a set C, where each codeword is an m × n
q-ary matrix, C = [ci,j], 0 ≤ i < m, 0 ≤ j < n.
The symbols c0,j, c1,j, . . . , c(m−1),j of the j-th column are
transmitted in this order over the j-th channel, Wj.
The RT-weight of C, denoted by wRT (C), of an m × n matrix is
the number of its elements that are nonzero or that lie below a
non-zero element.

Matrix codes - cont.
The RT-distance, denoted by dRT (C, C ), between the matrices C
and C of equal size is the number of positions in which the matrices
diﬀer or that lie below positions in which the matrices diﬀer.
The minimum RT-distance, dRT (C), of the matrix code C consisting
of at least two codewords, is the minimum RT-distance between any
two distinct codewords in C.
If m = 1, (minimum) RT-distance is the (minimum) Hamming
distance, and RT-weight is the Hamming weight.

Matrix codes - cont.
The RT-distance, denoted by dRT (C, C ), between the matrices C
and C of equal size is the number of positions in which the matrices
diﬀer or that lie below positions in which the matrices diﬀer.
The minimum RT-distance, dRT (C), of the matrix code C consisting
of at least two codewords, is the minimum RT-distance between any
two distinct codewords in C.
If m = 1, (minimum) RT-distance is the (minimum) Hamming
distance, and RT-weight is the Hamming weight.
Let C be a matrix code consisting of M codewords, each is an
m × n q-ary matrix, that has minimum RT-distance dRT . Then,
we say that C is an (m × n, M, dRT )q matrix code.

Example
Consider the two matrices




0 0 0
0 1 1
0 1 0
0 0 0



 ,




0 0 1
1 0 0
0 0 0
0 0 0



 .
The ﬁrst matrix has RT-weight 6 and the second has RT-weight 7.
The RT-distance between them is 10.

One-dimensional construction
A one-dimensional (N, M, dH)q code, where N = mn, can be used
as an (m × n, M, dRT )q, where dRT ≥ dH, by arranging each code-
word of length N into an m × n matrix.
We call this construction the one-dimensional construction.

One-dimensional construction
A one-dimensional (N, M, dH)q code, where N = mn, can be used
as an (m × n, M, dRT )q, where dRT ≥ dH, by arranging each code-
word of length N into an m × n matrix.
We call this construction the one-dimensional construction.
For one-dimensional code of length 12 and of minimum distance 6
over GF(2),
A non-linear code has only 24 codewords.
A linear code has 16 codewords.
Hence, the one-dimensional constructions yield (4 × 3, M, 6)2 codes
with M ≤ 24.

Example
Consider the linear binary matrix code that has 64 codewords
which are obtained as linear combinations of the six matrices:




1 1 1
0 0 0
0 0 0
0 0 0



 ,




1 1 1
0 1 1
0 0 0
0 0 0



 ,




1 0 0
0 0 0
0 1 1
0 0 0



 ,




1 1 1
1 0 0
1 1 1
0 1 1



 ,




1 1 1
1 1 0
0 0 0
0 0 0



 ,




1 0 1
0 0 0
1 1 0
0 0 0



 .
It can be shown that any nonzero linear combination of the above
six matrices has RT-weight at least 6.

Singleton bound
For any (m × n, M, dRT )q matrix code with M ≥ 2, we have
dRT ≤ mn − logq M + 1.
It is the Singleton bound for (m × n, M, dRT )q matrix codes.
Block codes that achieve equality in Singleton bound are called
maximum distance separable (MDS) codes.

Galois-Fourier transform (GFT)
Let a(x) = n−1
t=0 atxt be a polynomial over the ﬁnite ﬁeld Fq of
characteristic p such that n is a factor of q − 1.
n is coprime with p.
Let α be an element in Fq of order n. We form the sequence c =
(c0, c1, . . . , cn−1) by evaluating a(x) at consecutive powers of α,
namely, cj = a(αj) for 0 ≤ j < n, i.e.,
cj = a(αj
) =
n−1
t=0
atαjt
.
The sequence c is the Galois-Fourier transform of sequence a.

Galois-Fourier transform (GFT) - cont.
The sequence a can be retrieved by taking the inverse Galois-Fourier
transform of the sequence c using the relation
at = (n)−1
p
n−1
j=0
cjα−jt
.
The number of zeros in the sequence c is equal to the number of
distinct roots of a(x) in the set {αj : 0 ≤ j < n}.
Since n−1
j=0 (x−αj) = xn −1, the Hamming weight of c is given by
wH(c) = wH(GFT(a)) = n − deg(GCD(a(x), xn
− 1)).

Reed-Solomon codes
The Reed-Solomon code of length n and parameters z and t∗ over
Fq, denoted by RS(n; z, t∗)q, is the set of Galois-Fourier transforms
of all sequences of length n over Fq that have zeros in the z cyclically
consecutive positions t∗, (t∗ + 1)n, . . . , (t∗ + z − 1)n, modulo n, i.e.,
RS(n; z, t∗
)q = {GFT(a0, a1, . . . , an−1) : at ∈ Fq, 0 ≤ t < n,
at∗ = a(t∗+1)n
= · · ·= a(t∗+z−1)n
= 0}.
The Reed-Solomon code is an [n, n − z, z + 1]q code.

BCH codes
Let Fq be a subfield of Fq, which is the case iff q is a power of q .
From RS(n; z, t∗)q, we define the BCH code, BCH(n; z, t∗)q , of
length n over Fq to be the set of codewords in the Reed-Solomon
code, RS(n; z, t∗)q, that are over Fq , i.e.,
BCH(n; z, t∗
)q = RS(n; z, t∗
)q ∩ Fn
q
= {GFT(a0, a1, . . . , an−1) ∈ Fn
q : at ∈ Fq,
at∗ = a(t∗+1)n
= · · ·= a(t∗+z−1)n
= 0}.
dimension: ≤ n − z
minimum distance: ≥ z + 1

Conjugacy constraint
We can specify the sequences of length n over Fq which have Galois-
Fourier transforms confined over the subfield Fq rather than Fq.
For a sequence a = (a0, a1, . . . , an−1) ∈ Fn
q , its Galois-Fourier trans-
form, GFT(a), is over the subfield Fq if and only if the sequence a
satisfies the so-called q -ary conjugacy constraints:
aq
t = a(tq )n
.
The BCH code is the set of Galois-Fourier transforms of all sequences
over Fq with zeros in cyclically consecutive positions that satisfy the
q -ary conjugacy constraints.

Hasse derivative
Let a(x) = N−1
t=0 atxt be a polynomial over the finite field Fq of
characteristic p. For a nonnegative integer i, the i-th Hasse deriva-
tive, a[i](x), of a(x) is defined by
a[i]
(x) =
N−1
t=0
t
i
atxt−i
.

Hasse-Galois-Fourier transform
Now suppose that N = pvn where n is a factor of q − 1. Then, Fq
has an element α of order n.
Corresponding to the polynomial a(x) = N−1
t=0 atxt, form a pv × n
matrix, C = [ci,j]0≤i<pv,0≤j<n, where ci,j is the value of the i-th
Hasse derivative of a(x) evaluated at αj, i.e.,
ci,j = a[i]
(αj
) =
N−1
t=0
t
i
atα(t−i)j
.

Hasse-Galois-Fourier transform
Now suppose that N = pvn where n is a factor of q − 1. Then, Fq
has an element α of order n.
Corresponding to the polynomial a(x) = N−1
t=0 atxt, form a pv × n
matrix, C = [ci,j]0≤i<pv,0≤j<n, where ci,j is the value of the i-th
Hasse derivative of a(x) evaluated at αj, i.e.,
ci,j = a[i]
(αj
) =
N−1
t=0
t
i
atα(t−i)j
.
The matrix C = [ci,j], obtained by evaluating the polynomial a(x)
and its Hasse derivatives at consecutive powers of α, is the Hasse-
Galois-Fourier transform of the sequence a = (a0, a1, . . . , aN−1) of
length N. We write C = HGFT(a).

Theorem
Let a = HGFT−1
(C). Then,
ar+ pv = (n)−1
p
pv−1
i=0
n−1
j=0
(−1)i−r i
r
α(i−r−pv )j
ci,j,
for 0 ≤ r < pv, 0 ≤ < n.
wRT(C) = N − deg(GCD(a(x), xN − 1)).

Matrix Reed-Solomon codes
The matrix Reed-Solomon code of size pv ×n and parameters z and
t∗, where 1 ≤ z < pvn and 0 ≤ t∗ < pvn, over Fq, denoted by
RS(pv × n; z, t∗)q, is the set of Hasse-Galois-Fourier transforms of
all sequences of length N = pvn over Fq that have zeros in the z
cyclically consecutive positions t∗, (t∗ + 1)pvn, . . . , (t∗ + z − 1)pvn,
modulo pvn, i.e.,
RS(pv
× n; z, t∗
)q = {HGFT(a0, a1, . . . , apvn−1) : at ∈ Fq,
0 ≤ t < pv
n, at∗ = a(t∗+1)pvn
= · · ·= a(t∗+z−1)pvn
= 0}.
Matrix Reed-Solomon code is a [pv × n, pvn − z, z + 1]q code.

Example
We construct a matrix [4 × 3, 8, 5]4 Reed-Solomon code over F4,
RS(4 × 3; 4, 1)4.

Example
We construct a matrix [4 × 3, 8, 5]4 Reed-Solomon code over F4,
RS(4 × 3; 4, 1)4.
The codewords in this code are the 48 = 65, 536 matrices over F4
that are the Hasse-Galois-Fourier transforms of the 48 sequences
(a0, a1, . . . , a11) of length 12 over F4 for which a1 = a2 = a3 =
a4 = 0. The Hasse-Galois-Fourier transforms of the eight sequences
(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0),
(0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0),
(0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),
(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1),
form a basis of the code composed of the following eight codewords
over F4:

Example




1 1 1
0 0 0
0 0 0
0 0 0



 ,




1 α2 α
1 α α2
0 0 0
0 0 0



 ,




1 1 1
0 0 0
1 α α2
0 0 0



 ,




1 α α2
1 1 1
1 α2 α
1 α α2



 ,




1 α2 α
0 0 0
0 0 0
0 0 0



 ,




1 1 1
1 α2 α
0 0 0
0 0 0



 ,




1 α α2
0 0 0
1 α2 α
0 0 0



 ,




1 α2 α
1 α α2
1 1 1
1 α2 α



 .

Matrix BCH codes
The matrix BCH code of size pv ×n over Fq , denoted by BCH(pv ×
n; z, t∗)q , is the set of codewords in the above Reed-Solomon code,
RS(pv × n; z, t∗)q, that are over Fq .
The matrix BCH code is the subﬁeld subcode of the Reed-Solomon
code given by
BCH(pv
× n; z, t∗
)q = RS(pv
× n; z, t∗
)q ∩ Fpv×n
q
= {HGFT(a0, a1, . . . , apvn−1) ∈ Fpv×n
q :
aj ∈ Fq, 0 ≤ j < pv
n − 1,
at∗ = a(t∗+1)pvn
= · · ·= a(t∗+z−1)pvn
= 0}.
dimension: ≤ pvn − z
minimum RT-distance: ≥ z + 1

Theorem
Let a = (a0, a1, . . . , apvn−1) be a sequence over Fq of length n that
divides q − 1. Let u, 0 ≤ u < n, be the inverse of pv modulo n, i.e.,
pvu ≡ 1 mod n. Then, HGFT(a) is over Fq , a subﬁeld of Fq, if and
only if a satisﬁes the q -ary generalized conjugacy constraints
pv−1
r=0
r
i
aq
r+((i−r)u+ )npv =
pv−1
r=0
r
i
ar+((i−r)u+q )npv ,
for all 0 ≤ i < pv and all 0 ≤ < n.

Example
For pv = 4 and n = 3, the sequence a = (a0, a1, . . . , a11) over F4 is the
inverse Hasse-Galois-Fourier transform of a 4 × 3 matrix over F2 iﬀ
a2
0 + a2
9 + a2
6 + a2
3 = a0 + a9 + a6 + a3,
a2
4 + a2
1 + a2
10 + a2
7 = a8 + a5 + a2 + a11,
a2
1 + a2
7 = a1 + a7,
a2
5 + a2
11 = a9 + a3,
a2
2 + a2
11 = a2 + a11,
a2
6 + a2
3 = a10 + a7,
a2
3 = a3,
a2
7 = a11.

Example
We construct the BCH(4 × 3; 4, 1)2 code over F2 with designed
minimum RT-distance of 5, where pv = 4 and n = 3.
The codewords in the matrix BCH code are precisely all matrices
over F2 which are Hasse-Galois-Fourier transforms of sequences
a = (a0, a1, . . . , a11) over F4 for which a1 = a2 = a3 = a4 = 0.

Example
We construct the BCH(4 × 3; 4, 1)2 code over F2 with designed
minimum RT-distance of 5, where pv = 4 and n = 3.
The codewords in the matrix BCH code are precisely all matrices
over F2 which are Hasse-Galois-Fourier transforms of sequences
a = (a0, a1, . . . , a11) over F4 for which a1 = a2 = a3 = a4 = 0.
Combining with the conjugacy constraint, the sequence a over F4
has the form
a = (a0, 0, 0, 0, 0, a5, a6, a7, a5 + a6 + a7, a2
5 + a7, a2
6 + a7, a7)
such that both a7 and a0 + a2
5 + a6 are in F2.
dimension: 6
true minimum RT-distance: 6

Cyclic structure of matrix RS code
Theorem
Let [ci,j], 0 ≤ i < pv, 0 ≤ j < n, be the Hasse-Galois-Fourier
transform of the sequence (a0, a1, . . . , aN−1), where N = pvn. Then,
[α−ici,(j−1)n
], 0 ≤ i < pv, 0 ≤ j < n, is the Hasse-Galois-Fourier trans-
form of the sequence (a0, α−1a1, . . . , α−(N−1)aN−1).

Cyclic structure of matrix RS code
Theorem
Let [ci,j], 0 ≤ i < pv, 0 ≤ j < n, be the Hasse-Galois-Fourier
transform of the sequence (a0, a1, . . . , aN−1), where N = pvn. Then,
[α−ici,(j−1)n
], 0 ≤ i < pv, 0 ≤ j < n, is the Hasse-Galois-Fourier trans-
form of the sequence (a0, α−1a1, . . . , α−(N−1)aN−1).
Example
Consider the two matrices
C =




α 0 α2
1 1 1
1 α α
0 1 1



 and C =




α2 α 0
α2 α2 α2
α2 α α2
1 0 1



 .
The matrix C is the matrix cyclic shift of C.

Example
C =




α 0 α2
1 1 1
1 α α
0 1 1



 and C =




α2 α 0
α2 α2 α2
α2 α α2
1 0 1



 .
Notice that
C = HGFT(α, 0, 0, 0, 0, 1, α2
, 1, 0, 0, α, 1),
C = HGFT(α, 0, 0, 0, 0, α, α2
, α2
, 0, 0, 1, α),
and both are codewords in RS(4 × 3; 4, 1)4.
Matrix BCH codes may not have the matrix cyclic shift property.

Inner product
Deﬁnition
C, C =
m−1
i=0
n−1
j=0
ci,jcm−i−1,j.
The inner product allows for developing a MacWilliams-type
identity for matrix codes.

The dual code of a matrix Reed-Solomon code
Theorem
Let C be the RS(pv × n; z, t∗)q code. Then, C⊥ is the
RS(pv × n; pvn − z, (pv − t∗)(pvn))q code.

The dual code of a matrix Reed-Solomon code
Theorem
Let C be the RS(pv × n; z, t∗)q code. Then, C⊥ is the
RS(pv × n; pvn − z, (pv − t∗)(pvn))q code.
Example
The dual of RS(4 × 3; 4, 1)4 is RS(4 × 3; 8, 3)4 which consists of all
linear combinations of the four matrices




1 1 1
0 0 0
0 0 0
0 0 0



 ,




1 α α2
1 1 1
0 0 0
0 0 0



 ,




1 α2 α
0 0 0
1 1 1
0 0 0



 ,




1 α2 α
1 α α2
1 1 1
1 α2 α



 .

The dual code of a matrix BCH code
Let Fq be a subﬁeld of Fq and q = q s.
The trace of an element β ∈ Fq over Fq is given by
Trq (β) = β + βq
+ · · · β(q )s−1
,
which is an element in Fq .
The trace code of C is then
Trq (C) = {(Trq (c0), Trq (c1), . . . , Trq (cn−1)) : (c0, c1, . . . , cn−1) ∈ C}.
and it is a linear code over Fq .

The dual code of a matrix BCH code -cont.
The Delsarte Theorem states that
(C|Fq )⊥
= Trq (C⊥
),
i.e., the dual of the subﬁeld subcode of a code is the trace of the
dual of that code.
BCH(pv
× n; z, t∗
)⊥
q = Trq (RS(pv
× n; pv
n − z, (pv
− t∗
)pvn)q).

Example
We construct a [3 × 2, 4, 3]4 Reed Solomon matrix code over F4. The
codewords in this code are the 44 = 256 matrices over F4 that are the
Hasse-Galois-Fourier transforms of the 44 sequences (a0, a1, . . . , a5) of
length 6 over F4 for which a4 = a5 = 0. The Hasse-Galois-Fourier
transforms of the four sequences
(1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0),
(0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0),
form a basis of the code composed of the following four codewords over
F4:
1 1 1
0 0 0
,
1 α α2
1 1 1
,
1 α2 α
0 0 0
,
1 1 1
1 α2 α
.

Example
It can be checked that the dual code of the subﬁeld subcode has 2(6−2) =
16 codewords, which are
0 0 0
0 0 0
,
0 0 0
0 1 1
,
0 1 1
0 0 0
,
0 1 1
0 1 1
,
0 0 1
1 0 1
,
0 0 1
1 1 0
,
0 1 0
1 0 1
,
0 1 0
1 1 0
,
1 0 1
0 0 0
,
1 0 1
0 1 1
,
1 1 0
0 0 0
,
1 1 0
0 1 1
,
1 0 0
1 0 1
,
1 0 0
1 1 0
,
1 1 1
1 0 1
,
1 1 1
1 1 0
.

Example
On the other hand, the dual code of the [3 × 2, 4, 3]4 Reed-Solomon
matrix code has the following 16 codewords,
0 0 0
0 0 0
,
α2 α 1
0 0 0
,
1 α2 α
0 0 0
,
α 1 α2
0 0 0
,
1 α 0
α2 α 1
,
α 0 1
α2 α 1
,
0 1 α
α2 α 1
,
α2 α2 α2
α2 α 1
,
α α2 0
1 α2 α
,
1 1 1
1 α2 α
,
α2 0 α
1 α2 α
,
0 α α2
1 α2 α
,
α2 1 0
α 1 α2 ,
0 α2 1
α 1 α2 ,
α α α
α 1 α2 ,
1 0 α2
α 1 α2 .
Using the fact that Tr(1) = 0, Tr(α) = Tr(α2) = 1, we can see that the
dual of subﬁeld subcode of the [3 × 2, 4, 3]4 Reed-Solomon matrix code
is the trace of dual code of Reed Solomon matrix code.

Summary
We study the true burst-correcting capability of Fire codes.
We construct BCH matrix codes and study their properties.

Thank you!

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