Best Known (144−30, 144, s)-Nets in Base 4
(144−30, 144, 1052)-Net over F4 — Constructive and digital
Digital (114, 144, 1052)-net over F4, using
- trace code for nets [i] based on digital (6, 36, 263)-net over F256, using
- net from sequence [i] based on digital (6, 262)-sequence over F256, using
(144−30, 144, 4224)-Net over F4 — Digital
Digital (114, 144, 4224)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4144, 4224, F4, 30) (dual of [4224, 4080, 31]-code), using
- 111 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 16 times 0, 1, 26 times 0, 1, 41 times 0) [i] based on linear OA(4133, 4102, F4, 30) (dual of [4102, 3969, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [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(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(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- 111 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 10 times 0, 1, 16 times 0, 1, 26 times 0, 1, 41 times 0) [i] based on linear OA(4133, 4102, F4, 30) (dual of [4102, 3969, 31]-code), using
(144−30, 144, 1289487)-Net in Base 4 — Upper bound on s
There is no (114, 144, 1289488)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 497 323986 853404 896211 356538 583524 943563 780915 483946 689500 017196 171084 837052 395097 212575 > 4144 [i]