### Table IV: Putative extremal quantum-error-correcting codes ((n; K; d)) in which K is a power of 2.

1998

Cited by 105

### Table IV: Putative extremal quantum-error-correcting codes ((n, K, d)) in which K is a power of 2.

1998

Cited by 105

### Table 5: A modified array representation of a (7,5,3) Reed-Solomon code over GF(a55a106a103 ), which can correct binary burst erasures or detect binary burst errors of length up to 6.

"... In PAGE 9: ... Considered as a binary (21,15) code, it is easy to verify that this code reaches the Reiger Bound: it can correct all binary burst erasures of length up to 6 or detect all binary burst errors of length up to 6. Example 3 Modified (7,5,3) Reed-Solomon code The code shown in Table5 is the dual of the code shown in Table 4, thus it is an (7,5,3) MDS code in GF(a55 a103 ) with its binary burst erasure correction and burst error detection capability maximized: it can correct any binary burst erasures or detect burst errors of length up to 6.... In PAGE 9: ...Table 5: A modified array representation of a (7,5,3) Reed-Solomon code over GF(a55a106a103 ), which can correct binary burst erasures or detect binary burst errors of length up to 6. The 6 binary symbols in the first and the last columns in Table5 are parity check symbols and the other symbols are original information symbols, where a189 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a191 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a190 a192 a50a114a126a21a26a120a125 a103 a33a92a43a89a126a110a33 a130 a126a2a33 a130 a103 a33a92a25a46a126a134a33a92a25 a107 a33a92a188a106a126a2a33a92a188 a107 a50 a107 a26a120a125a46a126a2a33a92a125 a107 a33a118a43 a103 a33 a130 a107 a33a92a25 a107 a33a92a25 a103 a33a92a188a106a126 a50 a103 a26a120a125 a107 a33a92a125 a103 a33a118a43a89a126a110a33a118a43 a107 a33 a130 a126a134a33 a130 a107 a33a118a25 a103 a33a118a188 a103 a42a77a126a21a26a22a125a46a126a134a33a92a43 a107 a33 a130 a126a110a33a118a25 a103 a33a118a188 a107 a42 a107 a26a22a125 a107 a33a92a43a89a126a2a33 a130 a103 a33a118a25a83a126a134a33a118a25 a103 a33a118a188a106a126a110a33a118a188 a103 a42 a103 a26a22a125 a103 a33a92a43 a103 a33 a130 a107 a33 a130 a103 a33a92a25 a107 a33a92a188 a107 a33a92a188 a103 a91 Finally, we briefly discuss decoding complexity of such modified MDS codes. When such a mod- ified (a28a38a37a39a32 ) MDS code over GF(a50 a10 ) are used to correct a burst erasure of length a43 in GF(a50 ) (where... In PAGE 11: ...Table5 is an (7,5) MDS code correcting a76a21a26a166a35 error over GF(a55 a103 ), thus trivially as a (21,15) binary code over GF(2), it can correct burst errors of length up to 1. The upper bound on the length of all binary burst errors it can correct is 3 by the Reiger Bound, but 2 by the Fire Bound.... ..."

### Table IV: Putative extremal quantum-error-correcting codes ((n; K; d)) in which K is a power of 2.

### Table 2: Shortest fixed-length codes of given bit-per-symbol values for a selection of (d, k) constraints.

"... In PAGE 3: ... A number of (d, k) combinations and the length of the shortest existing fixed- length codes for a given ratio of a/N were computed. These are listed in Table2 , which may be used as an aid in construction of both state-dependent and state- independent codes. In the latter case the Table indicates a lower bound on the code-word length for a given bit- per-symbol value.... In PAGE 4: ... Example Consider (d, k) = (1, 3). D and D2 are given by 0100 D=[: : 1000 polo D2=l 1101 1100 0100- Table2 shows that a bit-per-symbol value of 3 represents over 90 percent of the channel capacity. Use of the search algorithm described in the subsection on fixed-length codes of minimum length indicates that the shortest existing code with this bit-per-symbol value has the parameters CY = 1, N = 2.... In PAGE 5: ... The shortest maximum word lengths for a number of (d, k) constraints and bit-per-symbol values were com- puted with the above technique and are shown in Table 3. A comparison of the variable-length codes with the fixed- length codes of Table2 shows that the extra degree of freedom in variable-length coding often yields a very significant decrease in word length. For example, the (d, k) = (4, 9) constraints, with a bit-per-symbol value of 3, result in fixed-length codes with a minimum of 9 bits per word.... In PAGE 6: ... In the latter case, this can be done after reception of the next word, so that at worst, code words corresponding to four bits must be correctly received in order to identify word ter- mination after an error in detection. Table2 shows that fixed-length coding with the same bit-per-symbol value as in the current example would require channel words of minimum length 27, each rep- ... ..."

### Table 5: Random coding BER, N=100, K=50. 100 samples.

2001

"... In PAGE 32: ... From Table 4 we see that when sufficient time is given (indicated by cases when both BBMB and BFMB solve all problems), the average running time of BFMB is never worse than BBMB and is sometimes better by a factor of 3-8. In Table5 we report the Bit Error Rate (BER) for the same problems and algorithms as in Table 4. BER is a standard measure used in the coding literature denoting the fraction... ..."

Cited by 19

### Table 4: Random coding, N=100, K=50, avg w =21. 100 samples.

2001

"... In PAGE 30: ... We also tried to run bucket elimination (BE) on this set of problems, but the induced width w was too large and BE failed to solve any problems. In Table4 , there are ve horizontal blocks, each corresponding to a di erent value of channel noise . Each block reports a distribution over the 95% accuracy range.... In PAGE 31: ... 100 samples. For example, looking at the third block in Table4 (corresponding to = 0:32), we see that MBE with i=2 (column 3) solved 45% of the problems exactly (opt 0:95), in 0.05 seconds on the average.... In PAGE 32: ... As in case of Max-CSP, if BFMB did not nish within the preset time bound, it returned the MBE assignment. From Table4 we see that when su cient time is given (indicated by cases when both BBMB and BFMB solve all problems), the average running time of BFMB is never worse than BBMB and is sometimes better by a factor of 3-8. In Table 5 we report the Bit Error Rate (BER) for the same problems and algorithms as in Table 4.... In PAGE 32: ... From Table 4 we see that when su cient time is given (indicated by cases when both BBMB and BFMB solve all problems), the average running time of BFMB is never worse than BBMB and is sometimes better by a factor of 3-8. In Table 5 we report the Bit Error Rate (BER) for the same problems and algorithms as in Table4 . BER is a standard measure used in the coding literature denoting the fraction... ..."

Cited by 19

### Table 4: Random coding, N=100, K=50, avg w =21. 100 samples.

2001

"... In PAGE 30: ... We also tried to run bucket elimination (BE) on this set of problems, but the induced width w was too large and BE failed to solve any problems. In Table4 , there are ve horizontal blocks, each corresponding to a di erent value of channel noise . Each block reports a distribution over the 95% accuracy range.... In PAGE 31: ...nd BFMB. Columns 3 through 6 report the results on various i-bounds. Column 7 reports results for IBP. For example, looking at the third block in Table4 (corresponding to = 0:32), we see that MBE with i=2 (column 3) solved 45% of the problems exactly (opt 0:95), in 0.05 seconds on the average.... In PAGE 32: ... As in case of Max-CSP, if BFMB did not nish within the preset time bound, it returned the MBE assignment. From Table4 we see that when su cient time is given (indicated by cases when both BBMB and BFMB solve all problems) the average running time of BFMB is never worse than BBMB and is sometimes better by a factor of 3-8. In Table 5 we report the Bit Error Rate (BER) for the same problems and algorithms as in Table 4.... ..."

Cited by 19

### TABLE I1 SOME TWO-LEVEL CYCLIC UEP CODES OF COMPOSITE LENGTH n k n, n, k, k, s, s, d dRCH Nonzeros

1998

Cited by 1

### Table 1. The input k0 and output n0 word sizes of the outer BCH code used by normal DVB-S2 frames. Also listed is the error correcting capability t of the BCH code.

"... In PAGE 12: ... Because of this, the length k0 of the input to the BCH encoder and the length k of the input to the LDPC encoder (which equals the length n0 of the output of the BCH encoder) are variable and depend on the rate r of the LDPC code. Normal frames can be encoded at eleven different code rates, as shown in Table1 . Short frames can be encoded at all the same code rates except for rate r = 9=10, which is not supported, as shown in Table 2.... ..."