Best Known (103, 136, s)-Nets in Base 3
(103, 136, 400)-Net over F3 — Constructive and digital
Digital (103, 136, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 34, 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
(103, 136, 715)-Net over F3 — Digital
Digital (103, 136, 715)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3136, 715, F3, 33) (dual of [715, 579, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, 755, F3, 33) (dual of [755, 619, 34]-code), using
- construction XX applied to C1 = C([724,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([724,28]) [i] based on
- linear OA(3127, 728, F3, 32) (dual of [728, 601, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,27}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3112, 728, F3, 29) (dual of [728, 616, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(3130, 728, F3, 33) (dual of [728, 598, 34]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,28}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3109, 728, F3, 28) (dual of [728, 619, 29]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,27], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(36, 24, F3, 3) (dual of [24, 18, 4]-code or 24-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(30, 3, F3, 0) (dual of [3, 3, 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([724,27]), C2 = C([0,28]), C3 = C1 + C2 = C([0,27]), and C∩ = C1 ∩ C2 = C([724,28]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3136, 755, F3, 33) (dual of [755, 619, 34]-code), using
(103, 136, 36060)-Net in Base 3 — Upper bound on s
There is no (103, 136, 36061)-net in base 3, because
- 1 times m-reduction [i] would yield (103, 135, 36061)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 25789 929663 877097 277041 462461 009601 916661 045323 065280 766317 044673 > 3135 [i]