Best Known (78, 78+19, s)-Nets in Base 4
(78, 78+19, 1076)-Net over F4 — Constructive and digital
Digital (78, 97, 1076)-net over F4, using
- 41 times duplication [i] based on digital (77, 96, 1076)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (11, 20, 48)-net over F4, using
- trace code for nets [i] based on digital (1, 10, 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
- trace code for nets [i] based on digital (1, 10, 24)-net over F16, using
- digital (57, 76, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 19, 257)-net over F256, using
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F256 with g(F) = 0 and N(F) ≥ 257, using
- the rational function field F256(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 256)-sequence over F256, using
- trace code for nets [i] based on digital (0, 19, 257)-net over F256, using
- digital (11, 20, 48)-net over F4, using
- (u, u+v)-construction [i] based on
(78, 78+19, 4660)-Net over F4 — Digital
Digital (78, 97, 4660)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(497, 4660, F4, 19) (dual of [4660, 4563, 20]-code), using
- 546 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 20 times 0, 1, 34 times 0, 1, 57 times 0, 1, 88 times 0, 1, 132 times 0, 1, 187 times 0) [i] based on linear OA(485, 4102, F4, 19) (dual of [4102, 4017, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(485, 4096, F4, 19) (dual of [4096, 4011, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(479, 4096, F4, 18) (dual of [4096, 4017, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 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(18) ⊂ Ce(17) [i] based on
- 546 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 5 times 0, 1, 11 times 0, 1, 20 times 0, 1, 34 times 0, 1, 57 times 0, 1, 88 times 0, 1, 132 times 0, 1, 187 times 0) [i] based on linear OA(485, 4102, F4, 19) (dual of [4102, 4017, 20]-code), using
(78, 78+19, 3652604)-Net in Base 4 — Upper bound on s
There is no (78, 97, 3652605)-net in base 4, because
- 1 times m-reduction [i] would yield (78, 96, 3652605)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6277 115701 954117 714068 833035 597283 711495 127685 492652 035744 > 496 [i]