Best Known (41−14, 41, s)-Nets in Base 8
(41−14, 41, 256)-Net over F8 — Constructive and digital
Digital (27, 41, 256)-net over F8, using
- 3 times m-reduction [i] based on digital (27, 44, 256)-net over F8, using
- trace code for nets [i] based on digital (5, 22, 128)-net over F64, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 5 and N(F) ≥ 128, using
- net from sequence [i] based on digital (5, 127)-sequence over F64, using
- trace code for nets [i] based on digital (5, 22, 128)-net over F64, using
(41−14, 41, 516)-Net in Base 8 — Constructive
(27, 41, 516)-net in base 8, using
- 81 times duplication [i] based on (26, 40, 516)-net in base 8, using
- base change [i] based on digital (16, 30, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 15, 258)-net over F256, using
- net from sequence [i] based on digital (1, 257)-sequence over F256, using
- trace code for nets [i] based on digital (1, 15, 258)-net over F256, using
- base change [i] based on digital (16, 30, 516)-net over F16, using
(41−14, 41, 598)-Net over F8 — Digital
Digital (27, 41, 598)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(841, 598, F8, 14) (dual of [598, 557, 15]-code), using
- 77 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 20 times 0, 1, 49 times 0) [i] based on linear OA(837, 517, F8, 14) (dual of [517, 480, 15]-code), using
- construction XX applied to C1 = C([510,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([510,12]) [i] based on
- linear OA(834, 511, F8, 13) (dual of [511, 477, 14]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [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(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(831, 511, F8, 12) (dual of [511, 480, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [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([510,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([510,12]) [i] based on
- 77 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 20 times 0, 1, 49 times 0) [i] based on linear OA(837, 517, F8, 14) (dual of [517, 480, 15]-code), using
(41−14, 41, 94043)-Net in Base 8 — Upper bound on s
There is no (27, 41, 94044)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 10 634147 810313 242653 907590 281065 406418 > 841 [i]