Best Known (193−39, 193, s)-Nets in Base 2
(193−39, 193, 260)-Net over F2 — Constructive and digital
Digital (154, 193, 260)-net over F2, using
- t-expansion [i] based on digital (153, 193, 260)-net over F2, using
- 3 times m-reduction [i] based on digital (153, 196, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 49, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 49, 65)-net over F16, using
- 3 times m-reduction [i] based on digital (153, 196, 260)-net over F2, using
(193−39, 193, 515)-Net over F2 — Digital
Digital (154, 193, 515)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2193, 515, F2, 2, 39) (dual of [(515, 2), 837, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 527, F2, 2, 39) (dual of [(527, 2), 861, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2193, 1054, F2, 39) (dual of [1054, 861, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(2193, 1055, F2, 39) (dual of [1055, 862, 40]-code), using
- construction XX applied to C1 = C([1019,32]), C2 = C([0,34]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([1019,34]) [i] based on
- linear OA(2181, 1023, F2, 37) (dual of [1023, 842, 38]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,32}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2166, 1023, F2, 35) (dual of [1023, 857, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(2186, 1023, F2, 39) (dual of [1023, 837, 40]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−4,−3,…,34}, and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(2161, 1023, F2, 33) (dual of [1023, 862, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(26, 26, F2, 3) (dual of [26, 20, 4]-code or 26-cap in PG(5,2)), using
- discarding factors / shortening the dual code based on linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- linear OA(21, 6, F2, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([1019,32]), C2 = C([0,34]), C3 = C1 + C2 = C([0,32]), and C∩ = C1 ∩ C2 = C([1019,34]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2193, 1055, F2, 39) (dual of [1055, 862, 40]-code), using
- OOA 2-folding [i] based on linear OA(2193, 1054, F2, 39) (dual of [1054, 861, 40]-code), using
- discarding factors / shortening the dual code based on linear OOA(2193, 527, F2, 2, 39) (dual of [(527, 2), 861, 40]-NRT-code), using
(193−39, 193, 8705)-Net in Base 2 — Upper bound on s
There is no (154, 193, 8706)-net in base 2, because
- 1 times m-reduction [i] would yield (154, 192, 8706)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6279 389136 522943 377146 295289 859047 890131 164068 889677 284480 > 2192 [i]