Best Known (123, 123+34, s)-Nets in Base 4
(123, 123+34, 1048)-Net over F4 — Constructive and digital
Digital (123, 157, 1048)-net over F4, using
- 41 times duplication [i] based on digital (122, 156, 1048)-net over F4, using
- trace code for nets [i] based on digital (5, 39, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
- trace code for nets [i] based on digital (5, 39, 262)-net over F256, using
(123, 123+34, 3646)-Net over F4 — Digital
Digital (123, 157, 3646)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4157, 3646, F4, 34) (dual of [3646, 3489, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(4157, 4121, F4, 34) (dual of [4121, 3964, 35]-code), using
- construction XX applied to Ce(33) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- linear OA(4151, 4096, F4, 34) (dual of [4096, 3945, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 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(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(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(33) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4157, 4121, F4, 34) (dual of [4121, 3964, 35]-code), using
(123, 123+34, 868996)-Net in Base 4 — Upper bound on s
There is no (123, 157, 868997)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 33374 874187 134228 863009 324333 748417 123934 840744 897395 764864 339685 595578 589632 417729 083103 228160 > 4157 [i]