Best Known (133−27, 133, s)-Nets in Base 4
(133−27, 133, 1056)-Net over F4 — Constructive and digital
Digital (106, 133, 1056)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (12, 25, 28)-net over F4, using
- net from sequence [i] based on digital (12, 27)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 12 and N(F) ≥ 28, using
- net from sequence [i] based on digital (12, 27)-sequence over F4, using
- digital (81, 108, 1028)-net over F4, using
- trace code for nets [i] based on digital (0, 27, 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, 27, 257)-net over F256, using
- digital (12, 25, 28)-net over F4, using
(133−27, 133, 4432)-Net over F4 — Digital
Digital (106, 133, 4432)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4133, 4432, F4, 27) (dual of [4432, 4299, 28]-code), using
- 318 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0, 1, 50 times 0, 1, 74 times 0, 1, 108 times 0) [i] based on linear OA(4121, 4102, F4, 27) (dual of [4102, 3981, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(4121, 4096, F4, 27) (dual of [4096, 3975, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4115, 4096, F4, 26) (dual of [4096, 3981, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(26) ⊂ Ce(25) [i] based on
- 318 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 4 times 0, 1, 7 times 0, 1, 12 times 0, 1, 20 times 0, 1, 32 times 0, 1, 50 times 0, 1, 74 times 0, 1, 108 times 0) [i] based on linear OA(4121, 4102, F4, 27) (dual of [4102, 3981, 28]-code), using
(133−27, 133, 2451927)-Net in Base 4 — Upper bound on s
There is no (106, 133, 2451928)-net in base 4, because
- 1 times m-reduction [i] would yield (106, 132, 2451928)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 29 642857 015849 266236 046461 987076 388706 302527 984496 437546 733497 799278 100645 027270 > 4132 [i]