Best Known (111, 111+31, s)-Nets in Base 4
(111, 111+31, 1044)-Net over F4 — Constructive and digital
Digital (111, 142, 1044)-net over F4, using
- 42 times duplication [i] based on digital (109, 140, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 35, 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, 35, 261)-net over F256, using
(111, 111+31, 3268)-Net over F4 — Digital
Digital (111, 142, 3268)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4142, 3268, F4, 31) (dual of [3268, 3126, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4142, 4112, F4, 31) (dual of [4112, 3970, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- linear OA(4139, 4096, F4, 31) (dual of [4096, 3957, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- 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(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(41, 14, F4, 1) (dual of [14, 13, 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, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(4142, 4112, F4, 31) (dual of [4112, 3970, 32]-code), using
(111, 111+31, 977246)-Net in Base 4 — Upper bound on s
There is no (111, 142, 977247)-net in base 4, because
- 1 times m-reduction [i] would yield (111, 141, 977247)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 7 770776 000374 288770 200646 060166 734210 089452 682333 570861 921271 909507 400469 901730 067716 > 4141 [i]