Best Known (125−27, 125, s)-Nets in Base 4
(125−27, 125, 1044)-Net over F4 — Constructive and digital
Digital (98, 125, 1044)-net over F4, using
- 41 times duplication [i] based on digital (97, 124, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 31, 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, 31, 261)-net over F256, using
(125−27, 125, 3267)-Net over F4 — Digital
Digital (98, 125, 3267)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4125, 3267, F4, 27) (dual of [3267, 3142, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4125, 4114, F4, 27) (dual of [4114, 3989, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4121, 4096, F4, 27) (dual of [4096, 3975, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 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(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(41, 15, F4, 1) (dual of [15, 14, 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
- linear OA(41, 3, F4, 1) (dual of [3, 2, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4125, 4114, F4, 27) (dual of [4114, 3989, 28]-code), using
(125−27, 125, 1044736)-Net in Base 4 — Upper bound on s
There is no (98, 125, 1044737)-net in base 4, because
- 1 times m-reduction [i] would yield (98, 124, 1044737)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 452 316594 837946 657279 540883 150931 467092 827774 136895 640299 284582 656784 728064 > 4124 [i]