Best Known (101, 101+37, s)-Nets in Base 4
(101, 101+37, 531)-Net over F4 — Constructive and digital
Digital (101, 138, 531)-net over F4, using
- 3 times m-reduction [i] based on digital (101, 141, 531)-net over F4, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 177)-net over F64, using
(101, 101+37, 1028)-Net over F4 — Digital
Digital (101, 138, 1028)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4138, 1028, F4, 37) (dual of [1028, 890, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(4138, 1032, F4, 37) (dual of [1032, 894, 38]-code), using
- construction XX applied to Ce(36) ⊂ Ce(34) ⊂ Ce(33) [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(4126, 1024, F4, 34) (dual of [1024, 898, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(41, 7, F4, 1) (dual of [7, 6, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(36) ⊂ Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4138, 1032, F4, 37) (dual of [1032, 894, 38]-code), using
(101, 101+37, 96225)-Net in Base 4 — Upper bound on s
There is no (101, 138, 96226)-net in base 4, because
- 1 times m-reduction [i] would yield (101, 137, 96226)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 30358 699778 764394 227294 070137 467877 500798 834737 856284 392928 373106 303803 553521 108256 > 4137 [i]