Best Known (201−45, 201, s)-Nets in Base 4
(201−45, 201, 1048)-Net over F4 — Constructive and digital
Digital (156, 201, 1048)-net over F4, using
- 41 times duplication [i] based on digital (155, 200, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 50, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 50, 262)-net over F256, using
(201−45, 201, 3519)-Net over F4 — Digital
Digital (156, 201, 3519)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4201, 3519, F4, 45) (dual of [3519, 3318, 46]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 4105, F4, 45) (dual of [4105, 3904, 46]-code), using
- construction XX applied to Ce(44) ⊂ Ce(42) ⊂ Ce(41) [i] based on
- linear OA(4199, 4096, F4, 45) (dual of [4096, 3897, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(4193, 4096, F4, 43) (dual of [4096, 3903, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(4187, 4096, F4, 42) (dual of [4096, 3909, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(44) ⊂ Ce(42) ⊂ Ce(41) [i] based on
- discarding factors / shortening the dual code based on linear OA(4201, 4105, F4, 45) (dual of [4105, 3904, 46]-code), using
(201−45, 201, 897420)-Net in Base 4 — Upper bound on s
There is no (156, 201, 897421)-net in base 4, because
- 1 times m-reduction [i] would yield (156, 200, 897421)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 2 582307 090973 379515 989496 599565 048720 904424 191647 730181 625089 620227 003893 805308 880154 384466 738844 403844 322699 457490 966496 > 4200 [i]