Best Known (8, 8+6, s)-Nets in Base 4
(8, 8+6, 48)-Net over F4 — Constructive and digital
Digital (8, 14, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 7, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
(8, 8+6, 65)-Net over F4 — Digital
Digital (8, 14, 65)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(414, 65, F4, 6) (dual of [65, 51, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(414, 70, F4, 6) (dual of [70, 56, 7]-code), using
- construction XX applied to C1 = C({0,1,2,47}), C2 = C([0,3]), C3 = C1 + C2 = C([0,2]), and C∩ = C1 ∩ C2 = C({0,1,2,3,47}) [i] based on
- linear OA(410, 63, F4, 4) (dual of [63, 53, 5]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,47}, and minimum distance d ≥ |{−1,0,1,2}|+1 = 5 (BCH-bound) [i]
- linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using the primitive cyclic code C(A) with length 63 = 43−1, defining set A = {0,1,2,3,47}, and minimum distance d ≥ |{−1,0,…,4}|+1 = 7 (BCH-bound) [i]
- linear OA(47, 63, F4, 3) (dual of [63, 56, 4]-code or 63-cap in PG(6,4)), using the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(40, 3, F4, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- construction XX applied to C1 = C({0,1,2,47}), C2 = C([0,3]), C3 = C1 + C2 = C([0,2]), and C∩ = C1 ∩ C2 = C({0,1,2,3,47}) [i] based on
- discarding factors / shortening the dual code based on linear OA(414, 70, F4, 6) (dual of [70, 56, 7]-code), using
(8, 8+6, 388)-Net in Base 4 — Upper bound on s
There is no (8, 14, 389)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 268 985332 > 414 [i]