Best Known (89, 89+30, s)-Nets in Base 4
(89, 89+30, 531)-Net over F4 — Constructive and digital
Digital (89, 119, 531)-net over F4, using
- 4 times m-reduction [i] based on digital (89, 123, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 7 and N(F) ≥ 177, using
- net from sequence [i] based on digital (7, 176)-sequence over F64, using
- trace code for nets [i] based on digital (7, 41, 177)-net over F64, using
(89, 89+30, 576)-Net in Base 4 — Constructive
(89, 119, 576)-net in base 4, using
- 1 times m-reduction [i] based on (89, 120, 576)-net in base 4, using
- trace code for nets [i] based on (9, 40, 192)-net in base 64, using
- 2 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F128 with g(F) = 3 and N(F) ≥ 192, using
- net from sequence [i] based on digital (3, 191)-sequence over F128, using
- base change [i] based on digital (3, 36, 192)-net over F128, using
- 2 times m-reduction [i] based on (9, 42, 192)-net in base 64, using
- trace code for nets [i] based on (9, 40, 192)-net in base 64, using
(89, 89+30, 1169)-Net over F4 — Digital
Digital (89, 119, 1169)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4119, 1169, F4, 30) (dual of [1169, 1050, 31]-code), using
- 132 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 23 times 0, 1, 35 times 0, 1, 44 times 0) [i] based on linear OA(4111, 1029, F4, 30) (dual of [1029, 918, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(28) [i] based on
- linear OA(4111, 1024, F4, 30) (dual of [1024, 913, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4106, 1024, F4, 29) (dual of [1024, 918, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(40, 5, F4, 0) (dual of [5, 5, 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
- 132 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 0, 1, 6 times 0, 1, 13 times 0, 1, 23 times 0, 1, 35 times 0, 1, 44 times 0) [i] based on linear OA(4111, 1029, F4, 30) (dual of [1029, 918, 31]-code), using
(89, 89+30, 127922)-Net in Base 4 — Upper bound on s
There is no (89, 119, 127923)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 441732 943651 614223 713864 262760 719078 596649 237607 121197 937333 658614 181696 > 4119 [i]