Best Known (143−37, 143, s)-Nets in Base 4
(143−37, 143, 531)-Net over F4 — Constructive and digital
Digital (106, 143, 531)-net over F4, using
- t-expansion [i] based on digital (105, 143, 531)-net over F4, using
- 4 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
- 4 times m-reduction [i] based on digital (105, 147, 531)-net over F4, using
(143−37, 143, 576)-Net in Base 4 — Constructive
(106, 143, 576)-net in base 4, using
- 1 times m-reduction [i] based on (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
- trace code for nets [i] based on (10, 48, 192)-net in base 64, using
(143−37, 143, 1191)-Net over F4 — Digital
Digital (106, 143, 1191)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4143, 1191, F4, 37) (dual of [1191, 1048, 38]-code), using
- 155 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 10 times 0, 1, 23 times 0, 1, 33 times 0, 1, 39 times 0, 1, 42 times 0) [i] based on linear OA(4137, 1030, F4, 37) (dual of [1030, 893, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 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(4131, 1024, F4, 35) (dual of [1024, 893, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- 155 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 10 times 0, 1, 23 times 0, 1, 33 times 0, 1, 39 times 0, 1, 42 times 0) [i] based on linear OA(4137, 1030, F4, 37) (dual of [1030, 893, 38]-code), using
(143−37, 143, 141432)-Net in Base 4 — Upper bound on s
There is no (106, 143, 141433)-net in base 4, because
- 1 times m-reduction [i] would yield (106, 142, 141433)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 31 085402 870351 474633 888087 566078 126543 863204 091755 347438 158441 478680 863749 983386 923980 > 4142 [i]