Best Known (58, 58+9, s)-Nets in Base 2
(58, 58+9, 16388)-Net over F2 — Constructive and digital
Digital (58, 67, 16388)-net over F2, using
- 21 times duplication [i] based on digital (57, 66, 16388)-net over F2, using
- net defined by OOA [i] based on linear OOA(266, 16388, F2, 9, 9) (dual of [(16388, 9), 147426, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(266, 16388, F2, 8, 9) (dual of [(16388, 8), 131038, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(266, 65553, F2, 9) (dual of [65553, 65487, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(249, 65536, F2, 7) (dual of [65536, 65487, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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 X applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(266, 65553, F2, 9) (dual of [65553, 65487, 10]-code), using
- appending kth column [i] based on linear OOA(266, 16388, F2, 8, 9) (dual of [(16388, 8), 131038, 10]-NRT-code), using
- net defined by OOA [i] based on linear OOA(266, 16388, F2, 9, 9) (dual of [(16388, 9), 147426, 10]-NRT-code), using
(58, 58+9, 18572)-Net over F2 — Digital
Digital (58, 67, 18572)-net over F2, using
- net defined by OOA [i] based on linear OOA(267, 18572, F2, 9, 9) (dual of [(18572, 9), 167081, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(267, 18572, F2, 8, 9) (dual of [(18572, 8), 148509, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(267, 18572, F2, 3, 9) (dual of [(18572, 3), 55649, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(267, 21851, F2, 3, 9) (dual of [(21851, 3), 65486, 10]-NRT-code), using
- 21 times duplication [i] based on linear OOA(266, 21851, F2, 3, 9) (dual of [(21851, 3), 65487, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(266, 65553, F2, 9) (dual of [65553, 65487, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(249, 65536, F2, 7) (dual of [65536, 65487, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 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 X applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 3-folding [i] based on linear OA(266, 65553, F2, 9) (dual of [65553, 65487, 10]-code), using
- 21 times duplication [i] based on linear OOA(266, 21851, F2, 3, 9) (dual of [(21851, 3), 65487, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(267, 21851, F2, 3, 9) (dual of [(21851, 3), 65486, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(267, 18572, F2, 3, 9) (dual of [(18572, 3), 55649, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(267, 18572, F2, 8, 9) (dual of [(18572, 8), 148509, 10]-NRT-code), using
(58, 58+9, 205133)-Net in Base 2 — Upper bound on s
There is no (58, 67, 205134)-net in base 2, because
- 1 times m-reduction [i] would yield (58, 66, 205134)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 73 788032 156389 252587 > 266 [i]