Best Known (73−15, 73, s)-Nets in Base 2
(73−15, 73, 152)-Net over F2 — Constructive and digital
Digital (58, 73, 152)-net over F2, using
- 21 times duplication [i] based on digital (57, 72, 152)-net over F2, using
- trace code for nets [i] based on digital (3, 18, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- trace code for nets [i] based on digital (3, 18, 38)-net over F16, using
(73−15, 73, 348)-Net over F2 — Digital
Digital (58, 73, 348)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(273, 348, F2, 3, 15) (dual of [(348, 3), 971, 16]-NRT-code), using
- OOA 3-folding [i] based on linear OA(273, 1044, F2, 15) (dual of [1044, 971, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(273, 1045, F2, 15) (dual of [1045, 972, 16]-code), using
- construction XX applied to C1 = C([1021,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1021,12]) [i] based on
- linear OA(261, 1023, F2, 13) (dual of [1023, 962, 14]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(261, 1023, F2, 13) (dual of [1023, 962, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(271, 1023, F2, 15) (dual of [1023, 952, 16]-code), using the primitive BCH-code C(I) with length 1023 = 210−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(251, 1023, F2, 11) (dual of [1023, 972, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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, 11, F2, 1) (dual of [11, 10, 2]-code) (see above)
- construction XX applied to C1 = C([1021,10]), C2 = C([0,12]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([1021,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(273, 1045, F2, 15) (dual of [1045, 972, 16]-code), using
- OOA 3-folding [i] based on linear OA(273, 1044, F2, 15) (dual of [1044, 971, 16]-code), using
(73−15, 73, 4209)-Net in Base 2 — Upper bound on s
There is no (58, 73, 4210)-net in base 2, because
- 1 times m-reduction [i] would yield (58, 72, 4210)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4728 780087 043097 685608 > 272 [i]