Best Known (12, 12+14, s)-Nets in Base 81
(12, 12+14, 264)-Net over F81 — Constructive and digital
Digital (12, 26, 264)-net over F81, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (0, 7, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81 (see above)
- digital (1, 15, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- digital (0, 4, 82)-net over F81, using
(12, 12+14, 477)-Net over F81 — Digital
Digital (12, 26, 477)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8126, 477, F81, 14) (dual of [477, 451, 15]-code), using
- 62 step Varšamov–Edel lengthening with (ri) = (1, 61 times 0) [i] based on linear OA(8125, 414, F81, 14) (dual of [414, 389, 15]-code), using
- construction XX applied to C1 = C([35,47]), C2 = C([34,46]), C3 = C1 + C2 = C([35,46]), and C∩ = C1 ∩ C2 = C([34,47]) [i] based on
- linear OA(8123, 410, F81, 13) (dual of [410, 387, 14]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {35,36,…,47}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(8123, 410, F81, 13) (dual of [410, 387, 14]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {34,35,…,46}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(8125, 410, F81, 14) (dual of [410, 385, 15]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {34,35,…,47}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(8121, 410, F81, 12) (dual of [410, 389, 13]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {35,36,…,46}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([35,47]), C2 = C([34,46]), C3 = C1 + C2 = C([35,46]), and C∩ = C1 ∩ C2 = C([34,47]) [i] based on
- 62 step Varšamov–Edel lengthening with (ri) = (1, 61 times 0) [i] based on linear OA(8125, 414, F81, 14) (dual of [414, 389, 15]-code), using
(12, 12+14, 518184)-Net in Base 81 — Upper bound on s
There is no (12, 26, 518185)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 41 745661 436169 520449 873709 440871 992107 504036 135601 > 8126 [i]