Claims
- 1. A method for time or transform domain decoding of RS coded received messages which corrects both errors and erasures in decoding of RS coded messages by the use of the Euclidean algorithm without separate computations of the errata locator polynomial .tau.(x) and the errata evaluator polynomial A(x) comprising the steps of
- (a) computing the transform of the received m-bit vector over the Galois Field GF(2.sup.m) from Eq. (2) for the initial syndrome computation, then calculate the erasure locator polynomial .LAMBDA.(x) and defined deg {.lambda.(x)}=v,
- (b) computing the Forney syndrome polynomial T(x) from Eq. (7),
- (c) applying the Euclidean algorithm to x.sup.d-1 and T(x) as given by Eq. (7) to determine the errata locator polynomial .tau.(x) and errata evaluator polynomial A(x), using as initial values of the Euclidean algorithm .tau..sub.0 (x)=.LAMBDA.(x), .tau..sub.-1 (x)=0, R.sub.-1 (x)=x.sup.d-1 and R.sub.0 (x)=T(x), wherein the recursion in Eq. (9) for R.sub.s (x) and .tau..sub.s (x) terminates when deg {R.sub.s (x)}.ltoreq. (d+v-3)/2 for the first time for some value s=s', and then compute .tau.(x) and A(x) from Eqs. (10a), and (10b), and
- (d) for time domain decoding, computing the error and erasure locations by finding the roots of the errata location polynomial .tau.(x) by Eq. (4c), or for transform domain decoding, determining the error and erasure locations by subjecting the errata locations polynomial .tau.(x) and the syndrome S(x) from the syndrome computaton in step (a) to a transform error pattern calculation which produces extended syndromes and inverting the sequence of the error pattern thus determined.
- 2. A method as defined by claim 1 wherein step (c), for v=d-1, .tau.(x) is set equal to A(x) and A(x) is set equal to T(x).
- 3. A method as defined by claim 1 or claim 2 for time domain decoding, where the last step (d) is comprised of
- (e) conducting polynomial evaluation of the errata evaluator polynomial A(x) and obtaining the values of polynomial A(x)=A(.alpha..sup.-(N-1)), . . . A(.alpha..sup.-2), A(.alpha..sup.-1), A(.alpha..sup.-0),
- (f) computing from the errata locator polynomial .tau.(x) the value (x.sup.b-1 .tau.'(x)).sup.-1, where b is a constant chosen for the design of the decoder for a specific RS code,
- (g) determining from the errata locator polynomial .tau.(x) the roots using a Chien search,
- (h) multiplying the result of step (e) with the result of step (f),
- (i) performing a logical AND between the polynomial results of step (h), and
- (j) adding the received message to the results of step (i) to produce a decoded output of the RS coded received message with errors and erasures corrected.
- 4. A method as defined by claim 1 or claim 2 for transform domain decoding, where the last step (d) is comprised of
- (e) calculating from the errata locator polynomial .tau.(x) and the syndromes from step (a), an extended syndrome S'(x) which constitutes a transform of the error pattern,
- (f) from the extended syndrome S'(x) produced in step (e) produce an inverse transform e.sub.j 's, and
- (g) adding the received message to the inverse transform e.sub.j 's of the transform error pattern to produce a decoded output.
- 5. Apparatus for time domain decoding of received RS coded messages, which apparatus corrects both errors and erasures in decoding of RS codes comprised of
- syndrome means for initial syndrome computation of RS coded received messages for producing an extended syndrome s(x),
- means for erasure locator power calculation of erasure location data received with RS coded received messages,
- means responsive to said syndrome s(x) of the initial received message for polynomial expansion of said initial syndrome to produce a Forney syndrome polynomial T(x),
- means for expansion of said erasure location power calculation to produce an erasure locator polynomial .LAMBDA.(x),
- means for counting the number v of input erasures from said received RS coded messages,
- means responsive to said counting means for generating the d+v-3/2 to produce a signal I, where d is a fixed number chosen for the designated RS code, and v is the number of erasure locations counted,
- means responsive to the signals T(x), .LAMBDA.(x) and I to produce an errata location polynomial .tau.(x) and errata evaluator polynomial A(x),
- polynomial evaluation means responsive to the errata evaluator polynomial A(x) for evaluation of said polynomial A(x) by producing the following values A(.alpha..sup.-(N-1)), . . . A(.alpha..sup.-2), A(.alpha..sup.-1), A(.alpha..sup.-0), where the multiplication of A(x) is in a Berlekamp multiplier with the multiplicand .alpha..sup.i fixed, where .alpha. is a primitive element in the Galois Field,
- value means responsive to said errata locator polynomial .tau.(x) for computing the value (x.sup.b-1 .tau.'(x)).sup.-1, where b is a constant chosen for a specific RS code,
- means for determining the roots of the errata locator polynomial .tau.(x) in order to determine the locations of errors and erasures,
- multiplication means responsive to said polynomial evaluation means and value means for forming resultant magnitudes,
- means for forming the logical AND of the result of said magnitude with said locations of errors and erasures, and
- means for adding said logical AND results to the original message received at said syndrome computation means to form a decoded output constituting a restored message.
- 6. Apparatus for transform domain decoding of received RS coded messages, said coded messages having a GF(2.sup.m), where N=2.sup.m is the length of the (N,I) RS code over GF(2.sup.m) with a minimum distance, d, which apparatus corrects both errors and erasures in decoding of received RS coded messages comprised of
- syndrome means for initial syndrome computation of said RS coded received messages for producing an extended syndrome s(x),
- means for erasure locator power calculation of erasure location data received with said RS coded received messages,
- means responsive to said syndrome s(x) of the initial received message for polynomial expansion of said initial syndrome to produce a Forney syndrome polynomial T(x),
- means for expansion of said erasure location power calculation to produce an erasure locator polynomial .LAMBDA.(x),
- meams for counting the number v of input erasures from said received RS coded messages,
- means for generating the d+v-3/2 to produce a signal I, where d is a designed RS vector distance number chosen for the designated code, and v is the number of erasure locations counted,
- means responsive to the signals T(x), .LAMBDA.(x) and I to produce an errata locator polynomial .tau.(x) and errata evaluator polynomial A(x),
- means responsive to said errata locator polynomial .tau.(x) and said syndrome S(x) of the initial received message for producing extended transforms S'(x) of the errata pattern .tau.(x),
- means for inverting the transform S'(x) to recover the errata vector e.sub.j 's, and
- combining the errata vector e.sub.j 's with the RS encoded received message in an adder to produce a decoded output of the RS coded received message with errors and erasures corrected.
- 7. Apparatus as defined in claim 6 wherein said means for producing extended transforms S'(x) of the errata pattern is comprised of
- a first switch for introducing symbols S.sub.1 through S.sub.n to n cells, where each cell comprises
- a register for storing one preloaded and distributed value of .tau.(x),
- a multiplier for multiplying each of said symbols S.sub.1 to S.sub.n by said one preloaded and distributed value of .tau.(x) they are introduced by said first switch,
- an adder for adding the product of said multiplier and accumulated products of a preceding cell except that the first cell n, where the cells are numbered in descending order from n to 1, which adds zero to said product, and
- a register for storing the output of said adder.
ORIGIN OF INVENTION
The invention described herein was made in the performance of work under a NASA contract, and is subject to the provisions of Public Law 95-517 (35 USC 202) in which the Contractor has elected to retain title.
US Referenced Citations (4)