Best Known (191, 191+54, s)-Nets in Base 4
(191, 191+54, 1056)-Net over F4 — Constructive and digital
Digital (191, 245, 1056)-net over F4, using
- 41 times duplication [i] based on digital (190, 244, 1056)-net over F4, using
- trace code for nets [i] based on digital (7, 61, 264)-net over F256, using
- net from sequence [i] based on digital (7, 263)-sequence over F256, using
- trace code for nets [i] based on digital (7, 61, 264)-net over F256, using
(191, 191+54, 4215)-Net over F4 — Digital
Digital (191, 245, 4215)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4245, 4215, F4, 54) (dual of [4215, 3970, 55]-code), using
- 109 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 9 times 0, 1, 30 times 0, 1, 65 times 0) [i] based on linear OA(4241, 4102, F4, 54) (dual of [4102, 3861, 55]-code), using
- construction X applied to Ce(53) ⊂ Ce(52) [i] based on
- linear OA(4241, 4096, F4, 54) (dual of [4096, 3855, 55]-code), using an extension Ce(53) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,53], and designed minimum distance d ≥ |I|+1 = 54 [i]
- linear OA(4235, 4096, F4, 53) (dual of [4096, 3861, 54]-code), using an extension Ce(52) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,52], and designed minimum distance d ≥ |I|+1 = 53 [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(53) ⊂ Ce(52) [i] based on
- 109 step Varšamov–Edel lengthening with (ri) = (1, 0, 1, 9 times 0, 1, 30 times 0, 1, 65 times 0) [i] based on linear OA(4241, 4102, F4, 54) (dual of [4102, 3861, 55]-code), using
(191, 191+54, 1057824)-Net in Base 4 — Upper bound on s
There is no (191, 245, 1057825)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 3196 747034 381889 103577 890493 975550 680331 507930 180102 708936 066077 990993 726539 865293 658685 290435 230893 215584 820331 820808 395054 633171 727521 025385 888160 > 4245 [i]