Best Known (112−27, 112, s)-Nets in Base 3
(112−27, 112, 400)-Net over F3 — Constructive and digital
Digital (85, 112, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 28, 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
(112−27, 112, 646)-Net over F3 — Digital
Digital (85, 112, 646)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3112, 646, F3, 27) (dual of [646, 534, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3112, 747, F3, 27) (dual of [747, 635, 28]-code), using
- construction XX applied to C1 = C([339,364]), C2 = C([342,365]), C3 = C1 + C2 = C([342,364]), and C∩ = C1 ∩ C2 = C([339,365]) [i] based on
- linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {339,340,…,364}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(397, 728, F3, 24) (dual of [728, 631, 25]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {342,343,…,365}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3109, 728, F3, 27) (dual of [728, 619, 28]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {339,340,…,365}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {342,343,…,364}, and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([339,364]), C2 = C([342,365]), C3 = C1 + C2 = C([342,364]), and C∩ = C1 ∩ C2 = C([339,365]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3112, 747, F3, 27) (dual of [747, 635, 28]-code), using
(112−27, 112, 33581)-Net in Base 3 — Upper bound on s
There is no (85, 112, 33582)-net in base 3, because
- 1 times m-reduction [i] would yield (85, 111, 33582)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 91326 257685 129737 378121 589370 515246 436564 033382 060261 > 3111 [i]