Best Known (150−31, 150, s)-Nets in Base 4
(150−31, 150, 1055)-Net over F4 — Constructive and digital
Digital (119, 150, 1055)-net over F4, using
- 41 times duplication [i] based on digital (118, 149, 1055)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (10, 25, 27)-net over F4, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 10 and N(F) ≥ 27, using
- net from sequence [i] based on digital (10, 26)-sequence over F4, using
- digital (93, 124, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 31, 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, 31, 257)-net over F256, using
- digital (10, 25, 27)-net over F4, using
- (u, u+v)-construction [i] based on
(150−31, 150, 4335)-Net over F4 — Digital
Digital (119, 150, 4335)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4150, 4335, F4, 31) (dual of [4335, 4185, 32]-code), using
- 222 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 21 times 0, 1, 35 times 0, 1, 53 times 0, 1, 78 times 0) [i] based on linear OA(4139, 4102, F4, 31) (dual of [4102, 3963, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(29) [i] based on
- linear OA(4139, 4096, F4, 31) (dual of [4096, 3957, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4133, 4096, F4, 30) (dual of [4096, 3963, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(30) ⊂ Ce(29) [i] based on
- 222 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 13 times 0, 1, 21 times 0, 1, 35 times 0, 1, 53 times 0, 1, 78 times 0) [i] based on linear OA(4139, 4102, F4, 31) (dual of [4102, 3963, 32]-code), using
(150−31, 150, 2046941)-Net in Base 4 — Upper bound on s
There is no (119, 150, 2046942)-net in base 4, because
- 1 times m-reduction [i] would yield (119, 149, 2046942)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 509261 044531 920167 686682 026523 134667 467025 770562 747468 656836 532745 658863 155079 079580 995592 > 4149 [i]