Best Known (49, 49+29, s)-Nets in Base 8
(49, 49+29, 354)-Net over F8 — Constructive and digital
Digital (49, 78, 354)-net over F8, using
- 6 times m-reduction [i] based on digital (49, 84, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 42, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 42, 177)-net over F64, using
(49, 49+29, 514)-Net in Base 8 — Constructive
(49, 78, 514)-net in base 8, using
- trace code for nets [i] based on (10, 39, 257)-net in base 64, using
- 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
- base change [i] based on digital (0, 30, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- base change [i] based on digital (0, 30, 257)-net over F256, using
- 1 times m-reduction [i] based on (10, 40, 257)-net in base 64, using
(49, 49+29, 547)-Net over F8 — Digital
Digital (49, 78, 547)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(878, 547, F8, 29) (dual of [547, 469, 30]-code), using
- 28 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 21 times 0) [i] based on linear OA(876, 517, F8, 29) (dual of [517, 441, 30]-code), using
- construction XX applied to C1 = C([510,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([510,27]) [i] based on
- linear OA(873, 511, F8, 28) (dual of [511, 438, 29]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,26}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(873, 511, F8, 28) (dual of [511, 438, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(876, 511, F8, 29) (dual of [511, 435, 30]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,27}, and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(870, 511, F8, 27) (dual of [511, 441, 28]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,26], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(80, 3, F8, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([510,26]), C2 = C([0,27]), C3 = C1 + C2 = C([0,26]), and C∩ = C1 ∩ C2 = C([510,27]) [i] based on
- 28 step Varšamov–Edel lengthening with (ri) = (1, 5 times 0, 1, 21 times 0) [i] based on linear OA(876, 517, F8, 29) (dual of [517, 441, 30]-code), using
(49, 49+29, 80040)-Net in Base 8 — Upper bound on s
There is no (49, 78, 80041)-net in base 8, because
- 1 times m-reduction [i] would yield (49, 77, 80041)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 3451 317525 414015 845459 208462 459851 940976 809449 463539 241448 395569 170032 > 877 [i]