Best Known (106, 106+38, s)-Nets in Base 4
(106, 106+38, 531)-Net over F4 — Constructive and digital
Digital (106, 144, 531)-net over F4, using
- t-expansion [i] based on digital (105, 144, 531)-net over F4, using
- 3 times m-reduction [i] based on digital (105, 147, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 49, 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, 49, 177)-net over F64, using
- 3 times m-reduction [i] based on digital (105, 147, 531)-net over F4, using
(106, 106+38, 576)-Net in Base 4 — Constructive
(106, 144, 576)-net in base 4, using
- trace code for nets [i] based on (10, 48, 192)-net in base 64, using
- 1 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
- base change [i] based on digital (3, 42, 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, 42, 192)-net over F128, using
- 1 times m-reduction [i] based on (10, 49, 192)-net in base 64, using
(106, 106+38, 1097)-Net over F4 — Digital
Digital (106, 144, 1097)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4144, 1097, F4, 38) (dual of [1097, 953, 39]-code), using
- 65 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 21 times 0, 1, 32 times 0) [i] based on linear OA(4141, 1029, F4, 38) (dual of [1029, 888, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(4141, 1024, F4, 38) (dual of [1024, 883, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4136, 1024, F4, 37) (dual of [1024, 888, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [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(37) ⊂ Ce(36) [i] based on
- 65 step Varšamov–Edel lengthening with (ri) = (1, 9 times 0, 1, 21 times 0, 1, 32 times 0) [i] based on linear OA(4141, 1029, F4, 38) (dual of [1029, 888, 39]-code), using
(106, 106+38, 96606)-Net in Base 4 — Upper bound on s
There is no (106, 144, 96607)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 497 372807 964835 025462 919365 643514 400123 376850 625344 687768 769480 698391 649437 006199 260632 > 4144 [i]