Best Known (157, 157+44, s)-Nets in Base 4
(157, 157+44, 1052)-Net over F4 — Constructive and digital
Digital (157, 201, 1052)-net over F4, using
- 41 times duplication [i] based on digital (156, 200, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 50, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
- trace code for nets [i] based on digital (6, 50, 263)-net over F256, using
(157, 157+44, 4018)-Net over F4 — Digital
Digital (157, 201, 4018)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4201, 4018, F4, 44) (dual of [4018, 3817, 45]-code), using
- discarding factors / shortening the dual code based on linear OA(4201, 4111, F4, 44) (dual of [4111, 3910, 45]-code), using
- construction XX applied to Ce(44) ⊂ Ce(41) ⊂ Ce(40) [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(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(4181, 4096, F4, 41) (dual of [4096, 3915, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(41, 14, F4, 1) (dual of [14, 13, 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(41) ⊂ Ce(40) [i] based on
- discarding factors / shortening the dual code based on linear OA(4201, 4111, F4, 44) (dual of [4111, 3910, 45]-code), using
(157, 157+44, 955790)-Net in Base 4 — Upper bound on s
There is no (157, 201, 955791)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 10 329133 405204 890276 911417 833074 220026 587771 290367 504041 667405 591096 531034 966992 131779 691173 525040 860657 386702 658084 768904 > 4201 [i]