Best Known (110−24, 110, s)-Nets in Base 4
(110−24, 110, 1040)-Net over F4 — Constructive and digital
Digital (86, 110, 1040)-net over F4, using
- 42 times duplication [i] based on digital (84, 108, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 27, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
- trace code for nets [i] based on digital (3, 27, 260)-net over F256, using
(110−24, 110, 2885)-Net over F4 — Digital
Digital (86, 110, 2885)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4110, 2885, F4, 24) (dual of [2885, 2775, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4110, 4109, F4, 24) (dual of [4109, 3999, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(21) [i] based on
- linear OA(4109, 4096, F4, 25) (dual of [4096, 3987, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(497, 4096, F4, 22) (dual of [4096, 3999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(24) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4110, 4109, F4, 24) (dual of [4109, 3999, 25]-code), using
(110−24, 110, 582259)-Net in Base 4 — Upper bound on s
There is no (86, 110, 582260)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 685025 951132 918695 722354 158652 921479 485351 623932 508925 255312 170780 > 4110 [i]