Best Known (31, 31+15, s)-Nets in Base 8
(31, 31+15, 354)-Net over F8 — Constructive and digital
Digital (31, 46, 354)-net over F8, using
- 2 times m-reduction [i] based on digital (31, 48, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 24, 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, 24, 177)-net over F64, using
(31, 31+15, 518)-Net in Base 8 — Constructive
(31, 46, 518)-net in base 8, using
- trace code for nets [i] based on (8, 23, 259)-net in base 64, using
- 1 times m-reduction [i] based on (8, 24, 259)-net in base 64, using
- base change [i] based on digital (2, 18, 259)-net over F256, using
- net from sequence [i] based on digital (2, 258)-sequence over F256, using
- base change [i] based on digital (2, 18, 259)-net over F256, using
- 1 times m-reduction [i] based on (8, 24, 259)-net in base 64, using
(31, 31+15, 809)-Net over F8 — Digital
Digital (31, 46, 809)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(846, 809, F8, 15) (dual of [809, 763, 16]-code), using
- 286 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 26 times 0, 1, 57 times 0, 1, 86 times 0, 1, 107 times 0) [i] based on linear OA(840, 517, F8, 15) (dual of [517, 477, 16]-code), using
- construction XX applied to C1 = C([510,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([510,13]) [i] based on
- linear OA(837, 511, F8, 14) (dual of [511, 474, 15]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(837, 511, F8, 14) (dual of [511, 474, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(840, 511, F8, 15) (dual of [511, 471, 16]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(834, 511, F8, 13) (dual of [511, 477, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [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([510,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([510,13]) [i] based on
- 286 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 26 times 0, 1, 57 times 0, 1, 86 times 0, 1, 107 times 0) [i] based on linear OA(840, 517, F8, 15) (dual of [517, 477, 16]-code), using
(31, 31+15, 308598)-Net in Base 8 — Upper bound on s
There is no (31, 46, 308599)-net in base 8, because
- 1 times m-reduction [i] would yield (31, 45, 308599)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 43556 878532 225454 552147 736978 021559 562632 > 845 [i]