Best Known (102, 102+37, s)-Nets in Base 4
(102, 102+37, 531)-Net over F4 — Constructive and digital
Digital (102, 139, 531)-net over F4, using
- t-expansion [i] based on digital (101, 139, 531)-net over F4, using
- 2 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
- 2 times m-reduction [i] based on digital (101, 141, 531)-net over F4, using
(102, 102+37, 1047)-Net over F4 — Digital
Digital (102, 139, 1047)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4139, 1047, F4, 37) (dual of [1047, 908, 38]-code), using
- 11 step Varšamov–Edel lengthening with (ri) = (1, 10 times 0) [i] based on linear OA(4138, 1035, F4, 37) (dual of [1035, 897, 38]-code), using
- construction XX applied to C1 = C([311,345]), C2 = C([309,343]), C3 = C1 + C2 = C([311,343]), and C∩ = C1 ∩ C2 = C([309,345]) [i] based on
- linear OA(4131, 1023, F4, 35) (dual of [1023, 892, 36]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {311,312,…,345}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(4131, 1023, F4, 35) (dual of [1023, 892, 36]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {309,310,…,343}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(4136, 1023, F4, 37) (dual of [1023, 887, 38]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {309,310,…,345}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(4126, 1023, F4, 33) (dual of [1023, 897, 34]-code), using the primitive BCH-code C(I) with length 1023 = 45−1, defining interval I = {311,312,…,343}, and designed minimum distance d ≥ |I|+1 = 34 [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
- linear OA(41, 6, F4, 1) (dual of [6, 5, 2]-code) (see above)
- construction XX applied to C1 = C([311,345]), C2 = C([309,343]), C3 = C1 + C2 = C([311,343]), and C∩ = C1 ∩ C2 = C([309,345]) [i] based on
- 11 step Varšamov–Edel lengthening with (ri) = (1, 10 times 0) [i] based on linear OA(4138, 1035, F4, 37) (dual of [1035, 897, 38]-code), using
(102, 102+37, 103930)-Net in Base 4 — Upper bound on s
There is no (102, 139, 103931)-net in base 4, because
- 1 times m-reduction [i] would yield (102, 138, 103931)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 121435 026846 898536 602923 462082 481728 460295 854488 699972 014698 360764 560399 968554 414617 > 4138 [i]