Best Known (109, 109+30, s)-Nets in Base 4
(109, 109+30, 1044)-Net over F4 — Constructive and digital
Digital (109, 139, 1044)-net over F4, using
- 1 times m-reduction [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
(109, 109+30, 3471)-Net over F4 — Digital
Digital (109, 139, 3471)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4139, 3471, F4, 30) (dual of [3471, 3332, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4139, 4121, F4, 30) (dual of [4121, 3982, 31]-code), using
- construction XX applied to Ce(29) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4133, 4096, F4, 30) (dual of [4096, 3963, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4115, 4096, F4, 26) (dual of [4096, 3981, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(45, 24, F4, 3) (dual of [24, 19, 4]-code or 24-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(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(29) ⊂ Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4139, 4121, F4, 30) (dual of [4121, 3982, 31]-code), using
(109, 109+30, 812322)-Net in Base 4 — Upper bound on s
There is no (109, 139, 812323)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 485676 054433 580178 837819 728481 736378 672857 718975 596029 737757 398151 157678 101307 087616 > 4139 [i]