Best Known (50−9, 50, s)-Nets in Base 2
(50−9, 50, 1027)-Net over F2 — Constructive and digital
Digital (41, 50, 1027)-net over F2, using
- net defined by OOA [i] based on linear OOA(250, 1027, F2, 9, 9) (dual of [(1027, 9), 9193, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(250, 1027, F2, 8, 9) (dual of [(1027, 8), 8166, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(250, 4109, F2, 9) (dual of [4109, 4059, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(237, 4096, F2, 7) (dual of [4096, 4059, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(21, 13, F2, 1) (dual of [13, 12, 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(250, 4109, F2, 9) (dual of [4109, 4059, 10]-code), using
- appending kth column [i] based on linear OOA(250, 1027, F2, 8, 9) (dual of [(1027, 8), 8166, 10]-NRT-code), using
(50−9, 50, 1370)-Net over F2 — Digital
Digital (41, 50, 1370)-net over F2, using
- net defined by OOA [i] based on linear OOA(250, 1370, F2, 9, 9) (dual of [(1370, 9), 12280, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(250, 1370, F2, 8, 9) (dual of [(1370, 8), 10910, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(250, 1370, F2, 3, 9) (dual of [(1370, 3), 4060, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- construction X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(249, 4096, F2, 9) (dual of [4096, 4047, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(237, 4096, F2, 7) (dual of [4096, 4059, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(213, 14, F2, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,2)), using
- dual of repetition code with length 14 [i]
- linear OA(21, 14, F2, 1) (dual of [14, 13, 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 X4 applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 3-folding [i] based on linear OA(250, 4110, F2, 9) (dual of [4110, 4060, 10]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(250, 1370, F2, 3, 9) (dual of [(1370, 3), 4060, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(250, 1370, F2, 8, 9) (dual of [(1370, 8), 10910, 10]-NRT-code), using
(50−9, 50, 10775)-Net in Base 2 — Upper bound on s
There is no (41, 50, 10776)-net in base 2, because
- 1 times m-reduction [i] would yield (41, 49, 10776)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 562 996012 071191 > 249 [i]