Best Known (71−32, 71, s)-Nets in Base 27
(71−32, 71, 196)-Net over F27 — Constructive and digital
Digital (39, 71, 196)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (4, 14, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (4, 20, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27 (see above)
- digital (5, 37, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (4, 14, 64)-net over F27, using
(71−32, 71, 370)-Net in Base 27 — Constructive
(39, 71, 370)-net in base 27, using
- 21 times m-reduction [i] based on (39, 92, 370)-net in base 27, using
- base change [i] based on digital (16, 69, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- base change [i] based on digital (16, 69, 370)-net over F81, using
(71−32, 71, 928)-Net over F27 — Digital
Digital (39, 71, 928)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2771, 928, F27, 32) (dual of [928, 857, 33]-code), using
- 185 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 28 times 0, 1, 50 times 0, 1, 74 times 0) [i] based on linear OA(2760, 732, F27, 32) (dual of [732, 672, 33]-code), using
- construction XX applied to C1 = C([727,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([727,30]) [i] based on
- linear OA(2758, 728, F27, 31) (dual of [728, 670, 32]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2758, 728, F27, 31) (dual of [728, 670, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2760, 728, F27, 32) (dual of [728, 668, 33]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2756, 728, F27, 30) (dual of [728, 672, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(270, 2, F27, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([727,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([727,30]) [i] based on
- 185 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 0, 0, 0, 1, 7 times 0, 1, 15 times 0, 1, 28 times 0, 1, 50 times 0, 1, 74 times 0) [i] based on linear OA(2760, 732, F27, 32) (dual of [732, 672, 33]-code), using
(71−32, 71, 587808)-Net in Base 27 — Upper bound on s
There is no (39, 71, 587809)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 423476 786462 667099 574350 491808 257103 441471 660199 808902 344797 323888 767878 554481 961757 585280 839465 655745 > 2771 [i]