Best Known (107, 107+29, s)-Nets in Base 4
(107, 107+29, 1048)-Net over F4 — Constructive and digital
Digital (107, 136, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 34, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
(107, 107+29, 3708)-Net over F4 — Digital
Digital (107, 136, 3708)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4136, 3708, F4, 29) (dual of [3708, 3572, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4136, 4126, F4, 29) (dual of [4126, 3990, 30]-code), using
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- 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(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(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(41, 4, F4, 1) (dual of [4, 3, 2]-code), using
- Reed–Solomon code RS(3,4) [i]
- construction XX applied to Ce(28) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4136, 4126, F4, 29) (dual of [4126, 3990, 30]-code), using
(107, 107+29, 1287986)-Net in Base 4 — Upper bound on s
There is no (107, 136, 1287987)-net in base 4, because
- 1 times m-reduction [i] would yield (107, 135, 1287987)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 1897 154244 196479 829831 593701 663630 784523 565834 261772 894470 385365 490869 282526 318412 > 4135 [i]