Best Known (53, 53+15, s)-Nets in Base 2
(53, 53+15, 132)-Net over F2 — Constructive and digital
Digital (53, 68, 132)-net over F2, using
- trace code for nets [i] based on digital (2, 17, 33)-net over F16, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 2 and N(F) ≥ 33, using
- net from sequence [i] based on digital (2, 32)-sequence over F16, using
(53, 53+15, 223)-Net over F2 — Digital
Digital (53, 68, 223)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(268, 223, F2, 2, 15) (dual of [(223, 2), 378, 16]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(268, 267, F2, 2, 15) (dual of [(267, 2), 466, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(268, 534, F2, 15) (dual of [534, 466, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(267, 533, F2, 15) (dual of [533, 466, 16]-code), using
- adding a parity check bit [i] based on linear OA(266, 532, F2, 14) (dual of [532, 466, 15]-code), using
- construction XX applied to C1 = C([509,10]), C2 = C([1,12]), C3 = C1 + C2 = C([1,10]), and C∩ = C1 ∩ C2 = C([509,12]) [i] based on
- linear OA(255, 511, F2, 13) (dual of [511, 456, 14]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,10}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(254, 511, F2, 12) (dual of [511, 457, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(264, 511, F2, 15) (dual of [511, 447, 16]-code), using the primitive BCH-code C(I) with length 511 = 29−1, defining interval I = {−2,−1,…,12}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(245, 511, F2, 10) (dual of [511, 466, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 511 = 29−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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, 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 (see above)
- construction XX applied to C1 = C([509,10]), C2 = C([1,12]), C3 = C1 + C2 = C([1,10]), and C∩ = C1 ∩ C2 = C([509,12]) [i] based on
- adding a parity check bit [i] based on linear OA(266, 532, F2, 14) (dual of [532, 466, 15]-code), using
- 1 times code embedding in larger space [i] based on linear OA(267, 533, F2, 15) (dual of [533, 466, 16]-code), using
- OOA 2-folding [i] based on linear OA(268, 534, F2, 15) (dual of [534, 466, 16]-code), using
- discarding factors / shortening the dual code based on linear OOA(268, 267, F2, 2, 15) (dual of [(267, 2), 466, 16]-NRT-code), using
(53, 53+15, 2561)-Net in Base 2 — Upper bound on s
There is no (53, 68, 2562)-net in base 2, because
- 1 times m-reduction [i] would yield (53, 67, 2562)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 147 722289 485177 644352 > 267 [i]