Best Known (64, 78, s)-Nets in Base 4
(64, 78, 2346)-Net over F4 — Constructive and digital
Digital (64, 78, 2346)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (57, 71, 2341)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 2341, F4, 14, 14) (dual of [(2341, 14), 32703, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(471, 16387, F4, 14) (dual of [16387, 16316, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(471, 16391, F4, 14) (dual of [16391, 16320, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(471, 16391, F4, 14) (dual of [16391, 16320, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(471, 16387, F4, 14) (dual of [16387, 16316, 15]-code), using
- net defined by OOA [i] based on linear OOA(471, 2341, F4, 14, 14) (dual of [(2341, 14), 32703, 15]-NRT-code), using
- digital (0, 7, 5)-net over F4, using
(64, 78, 12857)-Net over F4 — Digital
Digital (64, 78, 12857)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(478, 12857, F4, 14) (dual of [12857, 12779, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(478, 16392, F4, 14) (dual of [16392, 16314, 15]-code), using
- (u, u+v)-construction [i] based on
- linear OA(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- dual of repetition code with length 8 [i]
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using
- an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(47, 8, F4, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,4)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(478, 16392, F4, 14) (dual of [16392, 16314, 15]-code), using
(64, 78, 5760570)-Net in Base 4 — Upper bound on s
There is no (64, 78, 5760571)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 91343 951990 878670 290142 620920 478603 751821 125376 > 478 [i]