Best Known (109, 109+31, s)-Nets in Base 4
(109, 109+31, 1044)-Net over F4 — Constructive and digital
Digital (109, 140, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 35, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
(109, 109+31, 2968)-Net over F4 — Digital
Digital (109, 140, 2968)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4140, 2968, F4, 31) (dual of [2968, 2828, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4140, 4109, F4, 31) (dual of [4109, 3969, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [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(4127, 4096, F4, 29) (dual of [4096, 3969, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4140, 4109, F4, 31) (dual of [4109, 3969, 32]-code), using
(109, 109+31, 812322)-Net in Base 4 — Upper bound on s
There is no (109, 140, 812323)-net in base 4, because
- 1 times m-reduction [i] would yield (109, 139, 812323)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 485676 054433 580178 837819 728481 736378 672857 718975 596029 737757 398151 157678 101307 087616 > 4139 [i]