Best Known (27−14, 27, s)-Nets in Base 27
(27−14, 27, 112)-Net over F27 — Constructive and digital
Digital (13, 27, 112)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (2, 9, 48)-net over F27, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 2 and N(F) ≥ 48, using
- net from sequence [i] based on digital (2, 47)-sequence over F27, using
- digital (4, 18, 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 (2, 9, 48)-net over F27, using
(27−14, 27, 160)-Net in Base 27 — Constructive
(13, 27, 160)-net in base 27, using
- 5 times m-reduction [i] based on (13, 32, 160)-net in base 27, using
- base change [i] based on digital (5, 24, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- base change [i] based on digital (5, 24, 160)-net over F81, using
(27−14, 27, 333)-Net over F27 — Digital
Digital (13, 27, 333)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2727, 333, F27, 2, 14) (dual of [(333, 2), 639, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2727, 366, F27, 2, 14) (dual of [(366, 2), 705, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2727, 732, F27, 14) (dual of [732, 705, 15]-code), using
- construction XX applied to C1 = C([727,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([727,12]) [i] based on
- linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2725, 728, F27, 13) (dual of [728, 703, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2727, 728, F27, 14) (dual of [728, 701, 15]-code), using the primitive BCH-code C(I) with length 728 = 272−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2723, 728, F27, 12) (dual of [728, 705, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [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([727,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([727,12]) [i] based on
- OOA 2-folding [i] based on linear OA(2727, 732, F27, 14) (dual of [732, 705, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(2727, 366, F27, 2, 14) (dual of [(366, 2), 705, 15]-NRT-code), using
(27−14, 27, 43140)-Net in Base 27 — Upper bound on s
There is no (13, 27, 43141)-net in base 27, because
- the generalized Rao bound for nets shows that 27m ≥ 443 450058 431524 783615 883659 842310 602915 > 2727 [i]