Best Known (36, 55, s)-Nets in Base 8
(36, 55, 354)-Net over F8 — Constructive and digital
Digital (36, 55, 354)-net over F8, using
- 3 times m-reduction [i] based on digital (36, 58, 354)-net over F8, using
- trace code for nets [i] based on digital (7, 29, 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, 29, 177)-net over F64, using
(36, 55, 516)-Net in Base 8 — Constructive
(36, 55, 516)-net in base 8, using
- 1 times m-reduction [i] based on (36, 56, 516)-net in base 8, using
- base change [i] based on digital (22, 42, 516)-net over F16, using
- trace code for nets [i] based on digital (1, 21, 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, 21, 258)-net over F256, using
- base change [i] based on digital (22, 42, 516)-net over F16, using
(36, 55, 634)-Net over F8 — Digital
Digital (36, 55, 634)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(855, 634, F8, 19) (dual of [634, 579, 20]-code), using
- 111 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 14 times 0, 1, 32 times 0, 1, 54 times 0) [i] based on linear OA(849, 517, F8, 19) (dual of [517, 468, 20]-code), using
- construction XX applied to C1 = C([510,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([510,17]) [i] based on
- linear OA(846, 511, F8, 18) (dual of [511, 465, 19]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(846, 511, F8, 18) (dual of [511, 465, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(849, 511, F8, 19) (dual of [511, 462, 20]-code), using the primitive BCH-code C(I) with length 511 = 83−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(843, 511, F8, 17) (dual of [511, 468, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([510,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([510,17]) [i] based on
- 111 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 14 times 0, 1, 32 times 0, 1, 54 times 0) [i] based on linear OA(849, 517, F8, 19) (dual of [517, 468, 20]-code), using
(36, 55, 155302)-Net in Base 8 — Upper bound on s
There is no (36, 55, 155303)-net in base 8, because
- 1 times m-reduction [i] would yield (36, 54, 155303)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 5 846163 536041 034021 810236 272647 970487 547555 946952 > 854 [i]