Best Known (72, 72+21, s)-Nets in Base 2
(72, 72+21, 138)-Net over F2 — Constructive and digital
Digital (72, 93, 138)-net over F2, using
- trace code for nets [i] based on digital (10, 31, 46)-net over F8, using
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 9, N(F) = 45, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
(72, 72+21, 227)-Net over F2 — Digital
Digital (72, 93, 227)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(293, 227, F2, 2, 21) (dual of [(227, 2), 361, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(293, 265, F2, 2, 21) (dual of [(265, 2), 437, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(293, 530, F2, 21) (dual of [530, 437, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(293, 531, F2, 21) (dual of [531, 438, 22]-code), using
- construction XX applied to C1 = C([509,16]), C2 = C([0,18]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([509,18]) [i] based on
- linear OA(282, 511, F2, 19) (dual of [511, 429, 20]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,16}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(282, 511, F2, 19) (dual of [511, 429, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(291, 511, F2, 21) (dual of [511, 420, 22]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,18}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(273, 511, F2, 17) (dual of [511, 438, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(21, 10, F2, 1) (dual of [10, 9, 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
- linear OA(21, 10, F2, 1) (dual of [10, 9, 2]-code) (see above)
- construction XX applied to C1 = C([509,16]), C2 = C([0,18]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([509,18]) [i] based on
- discarding factors / shortening the dual code based on linear OA(293, 531, F2, 21) (dual of [531, 438, 22]-code), using
- OOA 2-folding [i] based on linear OA(293, 530, F2, 21) (dual of [530, 437, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(293, 265, F2, 2, 21) (dual of [(265, 2), 437, 22]-NRT-code), using
(72, 72+21, 2649)-Net in Base 2 — Upper bound on s
There is no (72, 93, 2650)-net in base 2, because
- 1 times m-reduction [i] would yield (72, 92, 2650)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4969 775813 088033 899844 941776 > 292 [i]