Best Known (89, 89+23, s)-Nets in Base 4
(89, 89+23, 1051)-Net over F4 — Constructive and digital
Digital (89, 112, 1051)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (9, 20, 23)-net over F4, using
- 3 times m-reduction [i] based on digital (9, 23, 23)-net over F4, using
- digital (69, 92, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 23, 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, 23, 257)-net over F256, using
- digital (9, 20, 23)-net over F4, using
(89, 89+23, 4206)-Net over F4 — Digital
Digital (89, 112, 4206)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4112, 4206, F4, 23) (dual of [4206, 4094, 24]-code), using
- 95 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 14 times 0, 1, 23 times 0, 1, 37 times 0) [i] based on linear OA(4103, 4102, F4, 23) (dual of [4102, 3999, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(4103, 4096, F4, 23) (dual of [4096, 3993, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(22) ⊂ Ce(21) [i] based on
- 95 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 14 times 0, 1, 23 times 0, 1, 37 times 0) [i] based on linear OA(4103, 4102, F4, 23) (dual of [4102, 3999, 24]-code), using
(89, 89+23, 1946361)-Net in Base 4 — Upper bound on s
There is no (89, 112, 1946362)-net in base 4, because
- 1 times m-reduction [i] would yield (89, 111, 1946362)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 6 739988 302346 834354 013450 886932 897997 500300 939945 235877 536115 294536 > 4111 [i]