Best Known (139, 139+18, s)-Nets in Base 2
(139, 139+18, 14565)-Net over F2 — Constructive and digital
Digital (139, 157, 14565)-net over F2, using
- 22 times duplication [i] based on digital (137, 155, 14565)-net over F2, using
- t-expansion [i] based on digital (136, 155, 14565)-net over F2, using
- net defined by OOA [i] based on linear OOA(2155, 14565, F2, 19, 19) (dual of [(14565, 19), 276580, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2155, 131086, F2, 19) (dual of [131086, 130931, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 131090, F2, 19) (dual of [131090, 130935, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- linear OA(2154, 131072, F2, 19) (dual of [131072, 130918, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2137, 131072, F2, 17) (dual of [131072, 130935, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(21, 18, F2, 1) (dual of [18, 17, 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(18) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(2155, 131090, F2, 19) (dual of [131090, 130935, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2155, 131086, F2, 19) (dual of [131086, 130931, 20]-code), using
- net defined by OOA [i] based on linear OOA(2155, 14565, F2, 19, 19) (dual of [(14565, 19), 276580, 20]-NRT-code), using
- t-expansion [i] based on digital (136, 155, 14565)-net over F2, using
(139, 139+18, 26218)-Net over F2 — Digital
Digital (139, 157, 26218)-net over F2, using
- 22 times duplication [i] based on digital (137, 155, 26218)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 26218, F2, 5, 18) (dual of [(26218, 5), 130935, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2155, 131090, F2, 18) (dual of [131090, 130935, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2155, 131091, F2, 18) (dual of [131091, 130936, 19]-code), using
- construction X4 applied to C([0,18]) ⊂ C([1,16]) [i] based on
- linear OA(2154, 131071, F2, 19) (dual of [131071, 130917, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2136, 131071, F2, 16) (dual of [131071, 130935, 17]-code), using the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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 C([0,18]) ⊂ C([1,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2155, 131091, F2, 18) (dual of [131091, 130936, 19]-code), using
- OOA 5-folding [i] based on linear OA(2155, 131090, F2, 18) (dual of [131090, 130935, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2155, 26218, F2, 5, 18) (dual of [(26218, 5), 130935, 19]-NRT-code), using
(139, 139+18, 739681)-Net in Base 2 — Upper bound on s
There is no (139, 157, 739682)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 182688 459765 926629 862860 206745 999527 950387 545404 > 2157 [i]