System and method having optimal, systematic q-ary codes for correcting all asymmetric and symmetric errors of limited magnitude
AssigneeState of Oregon
InventorsBella Bose, Noha Elarief
Life of this patent
Estimating the term…
Abstract
A computer-implemented method and computer program product comprising optimal, systematic q-ary codes for correcting all asymmetric and symmetric errors of limited magnitude are provided.
Claims8
A computer-implemented method for systematic decoding of an information vector for use in correction of all asymmetric errors of maximum magnitude l and executed by a processor, comprising:
- a) providing a q-ary encoded vector that includes an information vector having k received information digits, (x′k-1, x′k-2, . . . , x′0) and having received check symbols (c′r-1, c′r-2, . . . , c′0), where
- r
- ≥
- k
- ×
- log
-
- (
- l
- +
- 1
- )
- log
-
- ⌈
- q
- l
- +
- 1
- ⌉
- ;
- b) recovering the check symbols (cr-1, cr-2, . . . , c0) by rounding each received check symbol which is not a multiple of (l+1) upwards to the nearest multiple of (l+1);
- c) computing a, the value of
- (
- c
- r
- -
- 1
- l
- +
- 1
- ,
- c
- r
- -
- 2
- l
- +
- 1
- ,
- …
-
- ,
- c
- 0
- l
- +
- 1
- )
- as a vector over
- Z
- ⌈
- q
- l
- +
- 1
- ⌉
- as
- a
- =
- (
- c
- r
- -
- 1
- l
- +
- 1
- )
-
- ⌈
- q
- l
- +
- 1
- ⌉
- r
- -
- 1
- +
- (
- c
- r
- -
- 2
- l
- +
- 1
- )
-
- ⌈
- q
- l
- +
- 1
- ⌉
- r
- -
- 2
- +
- …
- +
- (
- c
- 0
- l
- +
- 1
- )
-
- ⌈
- q
- l
- +
- 1
- ⌉
- 0
- ;
- d) representing a in radix (l+1) number with k digits as y=(yk-1,yk-2, . . . ,y0);
- e) computing ei=(yi−x′i) mod(l+1);
- and
- f) computing the corrected codeword, (xk-1, xk-2, . . . , x0), as xi=x′i+ei, where i=0, 1, . . . , k-1.
A computer program product, comprising a non-transitory computer-readable medium having a computer readable program code embodied therein, said computer readable program code adapted to be executed to implement a method for decoding an information vector for use in systematic correction of all asymmetric errors of maximum magnitude, l, comprising:
- a) providing a q-ary encoded vector that includes an information vector having k received information digits, (x′k-1, x′k-2, . . . , x′0) and having received check symbols (c′r-1, c′r-2, . . . , c′0), where
- r
- ≥
- k
- ×
- log
-
- (
- l
- +
- 1
- )
- log
-
- ⌈
- q
- l
- +
- 1
- ⌉
- ;
- b) recovering the check symbols (cr-1, cr-2, . . . , c0) by rounding each received check symbol which is not a multiple of (l+1) upwards to the nearest multiple of (l+1);
- c) computing a, the value of
- (
- c
- r
- -
- 1
- l
- +
- 1
- ,
- c
- r
- -
- 2
- l
- +
- 1
- ,
- …
-
- ,
- c
- 0
- l
- +
- 1
- )
- as a vector over
- Z
- ⌈
- q
- l
- +
- 1
- ⌉
- as
- a
- =
- (
- c
- r
- -
- 1
- l
- +
- 1
- )
-
- ⌈
- q
- l
- +
- 1
- ⌉
- r
- -
- 1
- +
- (
- c
- r
- -
- 2
- l
- +
- 1
- )
-
- ⌈
- q
- l
- +
- 1
- ⌉
- r
- -
- 2
- +
- …
- +
- (
- c
- 0
- l
- +
- 1
- )
-
- ⌈
- q
- l
- +
- 1
- ⌉
- 0
- ;
- d) representing a in radix (l+1) number with k digits as y=(yk-1,yk-2, . . . ,y0);
- e) computing ei=(yi−x′i) mod(l+1);
- and
- f) computing the corrected codeword, (xk-1, xk-2, . . . , x0), as xi=x′i+ei, where i=0, 1, . . . , k-1.
A computer-implemented method for systematic decoding of an information vector for use in correction of all symmetric errors of maximum magnitude l and executed by a processor, comprising:
- a) providing a q-ary encoded vector that includes an information vector having k received information digits, (x′k-1, x′k-2, . . . , x′0) and having received check symbols (c′r-1, c′r-2, . . . , c′0), where
- r
- ≥
- k
- ×
- log
-
- (
- 2
-
- l
- +
- 1
- )
- log
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- ;
- b) recovering the check symbols by rounding each received check symbol (cr-1, cr-2, . . . , c0) which is not a multiple of (2l+1) either upwards or downwards to the nearest multiple of (2l+1), whichever is closer;
- c) computing a, the value of
- (
- c
- r
- -
- 1
- 2
-
- l
- +
- 1
- ,
- c
- r
- -
- 2
- 2
-
- l
- +
- 1
- ,
- …
-
- ,
- c
- 0
- 2
-
- l
- +
- 1
- )
- as a vector over
- Z
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- as
- a
- =
- (
- c
- r
- -
- 1
- 2
-
- l
- +
- 1
- )
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- r
- -
- 1
- +
- (
- c
- r
- -
- 2
- 2
-
- l
- +
- 1
- )
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- r
- -
- 2
- +
- …
- +
- (
- c
- 0
- 2
-
- l
- +
- 1
- )
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- 0
- ;
- d) representing a in radix (2l+1) number with k digits as y=(yk-1,yk-2, . . . ,y0);
- e) computing ei=(yi−x′i)mod(2l+1) such that −l≦ei≦l;
- and
- f) computing the corrected codeword, (xk-1, xk-2, . . . , x0), as xi=x′i+ei, where i=0, 1, . . . , k-1.
A computer program product, comprising a non-transitory computer-readable medium having a computer readable program code embodied therein, said computer readable program code adapted to be executed to implement a method for decoding an information vector for use in systematic correction of all symmetric errors of maximum magnitude, l, comprising:
- a) providing a q-ary encoded vector that includes an information vector having k received information digits, (x′k-1, x′k-2, . . . , x′0) and having received check symbols (c′r-1, c′r-2, . . . , c′0), where
- r
- ≥
- k
- ×
- log
-
- (
- 2
-
- l
- +
- 1
- )
- log
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- ;
- b) recovering the check symbols (cr-1, cr-2, . . . , c0) by rounding each received check symbol which is not a multiple of (2l+1) either upwards or downwards to the nearest multiple of (2l+1), whichever is closer;
- c) computing a, the value of
- (
- c
- r
- -
- 1
- 2
-
- l
- +
- 1
- ,
- c
- r
- -
- 2
- 2
-
- l
- +
- 1
- ,
- …
-
- ,
- c
- 0
- 2
-
- l
- +
- 1
- )
- as a vector over
- Z
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- as
- a
- =
- (
- c
- r
- -
- 1
- 2
-
- l
- +
- 1
- )
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- r
- -
- 1
- +
- (
- c
- r
- -
- 2
- 2
-
- l
- +
- 1
- )
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- r
- -
- 2
- +
- …
- +
- (
- c
- 0
- 2
-
- l
- +
- 1
- )
-
- ⌈
- q
- 2
-
- l
- +
- 1
- ⌉
- 0
- ;
- d) representing a in radix (2l+1) number with k digits as y=(yk-1,yk-2, . . . ,y0);
- e) computing ei=(yi−x′i)mod(2l+1) such that −l≦ei≦l;
- and
- computing the corrected codeword, (xk-1, xk-2, . . . , x0), as xi=x′i+ei, where i=0, 1, . . . , k-1.
Dependencies are inferred from the claim text.