Best Known (104−23, 104, s)-Nets in Base 4
(104−23, 104, 1040)-Net over F4 — Constructive and digital
Digital (81, 104, 1040)-net over F4, using
- trace code for nets [i] based on digital (3, 26, 260)-net over F256, using
- net from sequence [i] based on digital (3, 259)-sequence over F256, using
(104−23, 104, 2580)-Net over F4 — Digital
Digital (81, 104, 2580)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4104, 2580, F4, 23) (dual of [2580, 2476, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4104, 4109, F4, 23) (dual of [4109, 4005, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(4103, 4096, F4, 23) (dual of [4096, 3993, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(491, 4096, F4, 21) (dual of [4096, 4005, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4104, 4109, F4, 23) (dual of [4109, 4005, 24]-code), using
(104−23, 104, 710163)-Net in Base 4 — Upper bound on s
There is no (81, 104, 710164)-net in base 4, because
- 1 times m-reduction [i] would yield (81, 103, 710164)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 102 844617 426787 207635 365683 330920 490661 747294 165257 500970 278754 > 4103 [i]