Best Known (138−30, 138, s)-Nets in Base 4
(138−30, 138, 1044)-Net over F4 — Constructive and digital
Digital (108, 138, 1044)-net over F4, using
- 42 times duplication [i] based on digital (106, 136, 1044)-net over F4, using
- trace code for nets [i] based on digital (4, 34, 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, 34, 261)-net over F256, using
(138−30, 138, 3302)-Net over F4 — Digital
Digital (108, 138, 3302)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4138, 3302, F4, 30) (dual of [3302, 3164, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4138, 4119, F4, 30) (dual of [4119, 3981, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(25) [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(45, 23, F4, 3) (dual of [23, 18, 4]-code or 23-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
- construction X applied to Ce(29) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(4138, 4119, F4, 30) (dual of [4119, 3981, 31]-code), using
(138−30, 138, 740611)-Net in Base 4 — Upper bound on s
There is no (108, 138, 740612)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 121418 797806 577535 772088 339003 048060 459215 103679 039574 915343 341775 267374 536226 495040 > 4138 [i]