Best Known (110, 110+31, s)-Nets in Base 4
(110, 110+31, 1044)-Net over F4 — Constructive and digital
Digital (110, 141, 1044)-net over F4, using
- 41 times duplication [i] based on 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
- trace code for nets [i] based on digital (4, 35, 261)-net over F256, using
(110, 110+31, 3114)-Net over F4 — Digital
Digital (110, 141, 3114)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4141, 3114, F4, 31) (dual of [3114, 2973, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 4110, F4, 31) (dual of [4110, 3969, 32]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(4140, 4109, F4, 31) (dual of [4109, 3969, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4141, 4110, F4, 31) (dual of [4110, 3969, 32]-code), using
(110, 110+31, 890976)-Net in Base 4 — Upper bound on s
There is no (110, 141, 890977)-net in base 4, because
- 1 times m-reduction [i] would yield (110, 140, 890977)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1 942693 756222 916261 609746 354258 006534 198987 547019 389050 769696 717298 318356 827526 149024 > 4140 [i]