Best Known (127, 164, s)-Nets in Base 4
(127, 164, 1044)-Net over F4 — Constructive and digital
Digital (127, 164, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 41, 261)-net over F256, using
- net from sequence [i] based on digital (4, 260)-sequence over F256, using
(127, 164, 2924)-Net over F4 — Digital
Digital (127, 164, 2924)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4164, 2924, F4, 37) (dual of [2924, 2760, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(4163, 4096, F4, 37) (dual of [4096, 3933, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(4157, 4096, F4, 35) (dual of [4096, 3939, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(4164, 4103, F4, 37) (dual of [4103, 3939, 38]-code), using
(127, 164, 712832)-Net in Base 4 — Upper bound on s
There is no (127, 164, 712833)-net in base 4, because
- 1 times m-reduction [i] would yield (127, 163, 712833)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 136 703904 403429 184301 657652 496583 037528 685922 172595 166614 162344 023693 608796 487724 390334 539256 950080 > 4163 [i]