Fourier Codes
Fourier Codes R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS
July 2009
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Summary
1 Introduction
Fourier Number Theoretic Transform - FNTT Eigensequences of the FNTT 2 Fourier Codes
Code Construction The Code Parameters
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Summary
3 Error Control based on the FNTT Eigenstructure
Single Error Correction Double Error Correction
4 Final Remarks
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Fourier Number Theoretic Transform - FNTT
Introduction Fourier Number Theoretic Transform - FNTT The GF (p)-valued sequences x = (x0 , x1 , . . . , xN −1 ) and X = (X0 , X1 , . . . , XN −1 ), form an unitary FNTT pair when N −1 X √ Xk = ( N )−1 (modp) xn αkn n=0
and
N −1 X √ xn = ( N )−1 (modp) Xk α−kn k =0
where α ∈ GF (p) has multiplicative order N and N
p−1 2
≡ 1(mod p) .
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Eigensequences of the FNTT
Introduction
Eigensequences of the FNTT A sequence x is said to be an eigensequence of the FNTT, with associated eigenvalue λ ∈ GF (p 2 ), when it satisfies X = λx . The eigenvalues of the FNTT are the fourth roots of unity (±1, ±j ), where j 2 ≡ −1(mod p).
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Eigensequences of the FNTT
Introduction
Eigensequences of the FNTT Lemma 1: If x is an FNTT eigensequence, then it has even symmetry (i.e. xi = xN −i ) if λ ≡ ±1(mod p) and odd symmetry
(i.e. xi = −xN −i ) if λ ≡ ±j (mod p).
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes
FNTT Matrix
√ −1 F = ( N ) (modp)
1
1
1
...
1
1 1 .. . 1
α α2 .. .
α2 α4 .. .
... ... .. .
αN −1 α2(N −1) .. .
αN −1
α2(N −1)
. . . α(N −1)(N −1)
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes
Code Construction If x is an eigensequence of the linear transform F , then Fx = λx . ⇒ (F − λI )x = 0 ⇒ H ∗ = (F − λI )
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes Example: Constructing linear block codes from the FNTT of length N = 5, over GF (41). Consider α = 10, an element of order 5 in the √ given field, 5 ≡ 13(mod 41) and j ≡ 9(mod 41). From the transform
matrix F one obtains
19 − λ 19 19 19 26 − λ 14 F − λI = 19 14 6 −λ 17 26 19 19
6
17
19 17
19 6
26 6−λ
17 14
14
26 − λ
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes Considering H = [IN −k |Pλ ] and G = [−PλT |Ik ], we have the following
codes
F (5, 2)
F (5, 1)
H
H
+1
−1
1
0 0
34 34
= 0 0
1 0 0 1
0 40 ; G +1 = 40 0
1
0 0
0 12
0 = 0 0
1 0 0 1
h 0 40 ; G −1 = 29 1 0 40 1 40
0 0
"
7 0 7 1
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
1 1 0 0
1
0 1
1 1
#
i
Introduction
Fourier Codes
Code Construction
Fourier Codes
F (5, 1)
F (5, 1)
H +j
H −j
1 0 = 0 0
0 0 1 0 0 1 0 0
1 0 0 1 = 0 0 0 0
0 0 1 0
0 h 1 ; G +j = 0 0 31 1 10 0 0
0 h 1 ; G −j = 0 40 0 37 1 4 0 0
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
i
40 10 31 1
4 37 1
i
Introduction
Fourier Codes
The Code Parameters
The Code Parameters Code block length n: order N of the FNTT matrix; Dimension k : Multiplicity of the associated eigenvalue λ; N
Mult of 1
Mult of -1
Mult of -j
Mult of +j
4m 4m+1
m+1 m+1
m m
m m
m-1 m
4m+2 4m+3
m+1 m+1
m+1 m+1
m m+1
m m
The multiplicity of λ depends on the value of
√ N (modp).
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
The Code Parameters
The Code Parameters Proposition 1: Let H λ = [In−k |P] be the parity-check matrix of an F λ (n, k , d ) Fourier Code over GF (p). Then the submatrix P contains a secondary diagonal matrix Ds , of order k , with entries m, where m = p − 1 if λ = ±1(modp) and m = 1 if λ = ±j (modp). for λ = ±1, m = p − 1
for λ = ±j , m = 1
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
The Code Parameters
The Code Parameters The Minimum Distance Proposition 2 (An upper bound on the minimum distance): The minimum distance of a Fourier code F λ (n, k , d ) satisfies d ≤ n − 2k + 2.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
The Code Parameters
The Code Parameters The Minimum Distance Corollary: For F λ (n, k , d ), with λ = ±j , the code minimum distance satisfies d ≤ n − 2k .
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Fourier Codes
Introduction
Fourier Codes
The Code Parameters
Parameters of some Fourier codes N
k +1
d +1
k −1
d −1
k +j
d +j
k −j
d −j
3
1
3
1
3
-
-
1
2
4 5
2 2
2 3
1 1
4 5
1
4
1 1
2 4
6 7
2 2
4 5
2 2
4 5
1 1
4 6
1 2
4 4
8 9
3 3
4 3
2 2
4 6
1 2
6 6
2 2
4 6
10 11
3 3
6 7
3 3
6 7
2 2
6 8
2 3
6 6
12
4
4
3
6
3
4
2
6
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Error Control based on the FNTT Eigenstructure
Syndrome Computation
S = Fr − λI r = x + e is the received sequence S is zero if, and only if, r is a codeword
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Single Error Correction
Single Error Correction - The symmetric case r = (r0 , r1 , r2 , ri , . . . , rN −i . . . , rN −1 ) According to Definition 1, if N is odd, r0 must satisfy √ r0 = (λ N − 1)−1 (r1 + r2 + . . . + rN −1 ) r = (r0 , r1 , r2 , ri , . . . , rN /2 , . . . , rN −i . . . , rN −1 ) If N is even, the possible error occurred at position r0 or rN /2 . √ rN /2 = r0 (λ N − 1)(r1 + . . . + rN /2−1 + rN /2+1 + . . . + rN −1 ) R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Final Remarks
Error Control based on the FNTT Eigenstructure
Final Remarks
Double Error Correction
Double Error Correction - The nonsymmetric case
3 possible options: 1
Errors in symbols r0 and ri , i 6= 0
2
Errors ri and rN −i
3
Error in ri , i 6= 0 and in rj , j 6= N − i
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Double Error Correction
Decoding algorithms for double errors
Decoding algorithm 2: r = (r0 , r1 , r2 , ri , . . . , rN −i . . . , rN −1 ) √ PN −1 Make ri = 12 (λ N r0 − j =0
j 6=i,N −i
rj ) and rN −i = ri ;
If the sequence is an eigensequence, decoding is complete. Otherwise, more than two errors have occurred.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Double Error Correction
Decoding algorithm 2 - Example 1 Consider the double-error correcting code F 1 (7, 2, 5) over GF (19), with √ α = 7, 7 ≡ 6(mod 29), generator matrix G=
"
16 20
0 1
1 0
10 20
10 20
1 0
0 1
#
,
and received sequence r = (16, 2, 1, 10, 10, 1, 3). √ PN −1 ri = 12 (λ N r0 − j =0
j 6=i,N −i
rj ) =⇒ r (1) = (16, 0, 1, 10, 10, 1, 0) and
R (1) = (16, 0, 1, 10, 10, 1, 0). =⇒ Decoding is complete!
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Final Remarks
We have introduced a new family of nonbinary linear block codes, the Fourier codes. The codewords of a Fourier code F λ (n, k , d ) are the eigensequences of the Fourier number theoretic transform, associated with a given eigenvalue λ.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Final Remarks
Strategies for single and double error control based on the FNTT eigenstructure were examined. The approach described can be extended to other families of finite field transforms The restrictions on the code parameters can be removed if arbitrary linear transforms are considered.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
July 2009
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Summary
1 Introduction
Fourier Number Theoretic Transform - FNTT Eigensequences of the FNTT 2 Fourier Codes
Code Construction The Code Parameters
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Summary
3 Error Control based on the FNTT Eigenstructure
Single Error Correction Double Error Correction
4 Final Remarks
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Fourier Number Theoretic Transform - FNTT
Introduction Fourier Number Theoretic Transform - FNTT The GF (p)-valued sequences x = (x0 , x1 , . . . , xN −1 ) and X = (X0 , X1 , . . . , XN −1 ), form an unitary FNTT pair when N −1 X √ Xk = ( N )−1 (modp) xn αkn n=0
and
N −1 X √ xn = ( N )−1 (modp) Xk α−kn k =0
where α ∈ GF (p) has multiplicative order N and N
p−1 2
≡ 1(mod p) .
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Eigensequences of the FNTT
Introduction
Eigensequences of the FNTT A sequence x is said to be an eigensequence of the FNTT, with associated eigenvalue λ ∈ GF (p 2 ), when it satisfies X = λx . The eigenvalues of the FNTT are the fourth roots of unity (±1, ±j ), where j 2 ≡ −1(mod p).
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Eigensequences of the FNTT
Introduction
Eigensequences of the FNTT Lemma 1: If x is an FNTT eigensequence, then it has even symmetry (i.e. xi = xN −i ) if λ ≡ ±1(mod p) and odd symmetry
(i.e. xi = −xN −i ) if λ ≡ ±j (mod p).
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes
FNTT Matrix
√ −1 F = ( N ) (modp)
1
1
1
...
1
1 1 .. . 1
α α2 .. .
α2 α4 .. .
... ... .. .
αN −1 α2(N −1) .. .
αN −1
α2(N −1)
. . . α(N −1)(N −1)
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes
Code Construction If x is an eigensequence of the linear transform F , then Fx = λx . ⇒ (F − λI )x = 0 ⇒ H ∗ = (F − λI )
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes Example: Constructing linear block codes from the FNTT of length N = 5, over GF (41). Consider α = 10, an element of order 5 in the √ given field, 5 ≡ 13(mod 41) and j ≡ 9(mod 41). From the transform
matrix F one obtains
19 − λ 19 19 19 26 − λ 14 F − λI = 19 14 6 −λ 17 26 19 19
6
17
19 17
19 6
26 6−λ
17 14
14
26 − λ
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
Code Construction
Fourier Codes Considering H = [IN −k |Pλ ] and G = [−PλT |Ik ], we have the following
codes
F (5, 2)
F (5, 1)
H
H
+1
−1
1
0 0
34 34
= 0 0
1 0 0 1
0 40 ; G +1 = 40 0
1
0 0
0 12
0 = 0 0
1 0 0 1
h 0 40 ; G −1 = 29 1 0 40 1 40
0 0
"
7 0 7 1
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
1 1 0 0
1
0 1
1 1
#
i
Introduction
Fourier Codes
Code Construction
Fourier Codes
F (5, 1)
F (5, 1)
H +j
H −j
1 0 = 0 0
0 0 1 0 0 1 0 0
1 0 0 1 = 0 0 0 0
0 0 1 0
0 h 1 ; G +j = 0 0 31 1 10 0 0
0 h 1 ; G −j = 0 40 0 37 1 4 0 0
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
i
40 10 31 1
4 37 1
i
Introduction
Fourier Codes
The Code Parameters
The Code Parameters Code block length n: order N of the FNTT matrix; Dimension k : Multiplicity of the associated eigenvalue λ; N
Mult of 1
Mult of -1
Mult of -j
Mult of +j
4m 4m+1
m+1 m+1
m m
m m
m-1 m
4m+2 4m+3
m+1 m+1
m+1 m+1
m m+1
m m
The multiplicity of λ depends on the value of
√ N (modp).
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
The Code Parameters
The Code Parameters Proposition 1: Let H λ = [In−k |P] be the parity-check matrix of an F λ (n, k , d ) Fourier Code over GF (p). Then the submatrix P contains a secondary diagonal matrix Ds , of order k , with entries m, where m = p − 1 if λ = ±1(modp) and m = 1 if λ = ±j (modp). for λ = ±1, m = p − 1
for λ = ±j , m = 1
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
Fourier Codes
The Code Parameters
The Code Parameters The Minimum Distance Proposition 2 (An upper bound on the minimum distance): The minimum distance of a Fourier code F λ (n, k , d ) satisfies d ≤ n − 2k + 2.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Introduction
The Code Parameters
The Code Parameters The Minimum Distance Corollary: For F λ (n, k , d ), with λ = ±j , the code minimum distance satisfies d ≤ n − 2k .
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Fourier Codes
Introduction
Fourier Codes
The Code Parameters
Parameters of some Fourier codes N
k +1
d +1
k −1
d −1
k +j
d +j
k −j
d −j
3
1
3
1
3
-
-
1
2
4 5
2 2
2 3
1 1
4 5
1
4
1 1
2 4
6 7
2 2
4 5
2 2
4 5
1 1
4 6
1 2
4 4
8 9
3 3
4 3
2 2
4 6
1 2
6 6
2 2
4 6
10 11
3 3
6 7
3 3
6 7
2 2
6 8
2 3
6 6
12
4
4
3
6
3
4
2
6
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Error Control based on the FNTT Eigenstructure
Syndrome Computation
S = Fr − λI r = x + e is the received sequence S is zero if, and only if, r is a codeword
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Single Error Correction
Single Error Correction - The symmetric case r = (r0 , r1 , r2 , ri , . . . , rN −i . . . , rN −1 ) According to Definition 1, if N is odd, r0 must satisfy √ r0 = (λ N − 1)−1 (r1 + r2 + . . . + rN −1 ) r = (r0 , r1 , r2 , ri , . . . , rN /2 , . . . , rN −i . . . , rN −1 ) If N is even, the possible error occurred at position r0 or rN /2 . √ rN /2 = r0 (λ N − 1)(r1 + . . . + rN /2−1 + rN /2+1 + . . . + rN −1 ) R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Final Remarks
Error Control based on the FNTT Eigenstructure
Final Remarks
Double Error Correction
Double Error Correction - The nonsymmetric case
3 possible options: 1
Errors in symbols r0 and ri , i 6= 0
2
Errors ri and rN −i
3
Error in ri , i 6= 0 and in rj , j 6= N − i
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Double Error Correction
Decoding algorithms for double errors
Decoding algorithm 2: r = (r0 , r1 , r2 , ri , . . . , rN −i . . . , rN −1 ) √ PN −1 Make ri = 12 (λ N r0 − j =0
j 6=i,N −i
rj ) and rN −i = ri ;
If the sequence is an eigensequence, decoding is complete. Otherwise, more than two errors have occurred.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Double Error Correction
Decoding algorithm 2 - Example 1 Consider the double-error correcting code F 1 (7, 2, 5) over GF (19), with √ α = 7, 7 ≡ 6(mod 29), generator matrix G=
"
16 20
0 1
1 0
10 20
10 20
1 0
0 1
#
,
and received sequence r = (16, 2, 1, 10, 10, 1, 3). √ PN −1 ri = 12 (λ N r0 − j =0
j 6=i,N −i
rj ) =⇒ r (1) = (16, 0, 1, 10, 10, 1, 0) and
R (1) = (16, 0, 1, 10, 10, 1, 0). =⇒ Decoding is complete!
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Final Remarks
We have introduced a new family of nonbinary linear block codes, the Fourier codes. The codewords of a Fourier code F λ (n, k , d ) are the eigensequences of the Fourier number theoretic transform, associated with a given eigenvalue λ.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
Error Control based on the FNTT Eigenstructure
Final Remarks
Final Remarks
Strategies for single and double error control based on the FNTT eigenstructure were examined. The approach described can be extended to other families of finite field transforms The restrictions on the code parameters can be removed if arbitrary linear transforms are considered.
R. M. Campello de Souza E. S. V. Freire H. M. de Oliveira Federal University of Pernambuco - UFPE Department of Electronic and Systems - DES Signal Processing Group - GPS Fourier Codes
