Best Known (123, 123+16, s)-Nets in Base 2
(123, 123+16, 16386)-Net over F2 — Constructive and digital
Digital (123, 139, 16386)-net over F2, using
- 21 times duplication [i] based on digital (122, 138, 16386)-net over F2, using
- t-expansion [i] based on digital (121, 138, 16386)-net over F2, using
- net defined by OOA [i] based on linear OOA(2138, 16386, F2, 17, 17) (dual of [(16386, 17), 278424, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2138, 131089, F2, 17) (dual of [131089, 130951, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2138, 131090, F2, 17) (dual of [131090, 130952, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(14) [i] based on
- 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(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(16) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(2138, 131090, F2, 17) (dual of [131090, 130952, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2138, 131089, F2, 17) (dual of [131089, 130951, 18]-code), using
- net defined by OOA [i] based on linear OOA(2138, 16386, F2, 17, 17) (dual of [(16386, 17), 278424, 18]-NRT-code), using
- t-expansion [i] based on digital (121, 138, 16386)-net over F2, using
(123, 123+16, 26218)-Net over F2 — Digital
Digital (123, 139, 26218)-net over F2, using
- 22 times duplication [i] based on digital (121, 137, 26218)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 26218, F2, 5, 16) (dual of [(26218, 5), 130953, 17]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2137, 131090, F2, 16) (dual of [131090, 130953, 17]-code), using
- 1 times truncation [i] based on linear OA(2138, 131091, F2, 17) (dual of [131091, 130953, 18]-code), using
- construction X4 applied to Ce(16) ⊂ Ce(14) [i] based on
- 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(2120, 131072, F2, 15) (dual of [131072, 130952, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 131071 = 217−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(218, 19, F2, 17) (dual of [19, 1, 18]-code), using
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- dual of repetition code with length 19 [i]
- strength reduction [i] based on linear OA(218, 19, F2, 18) (dual of [19, 1, 19]-code or 19-arc in PG(17,2)), using
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(16) ⊂ Ce(14) [i] based on
- 1 times truncation [i] based on linear OA(2138, 131091, F2, 17) (dual of [131091, 130953, 18]-code), using
- OOA 5-folding [i] based on linear OA(2137, 131090, F2, 16) (dual of [131090, 130953, 17]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 26218, F2, 5, 16) (dual of [(26218, 5), 130953, 17]-NRT-code), using
(123, 123+16, 639850)-Net in Base 2 — Upper bound on s
There is no (123, 139, 639851)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 696903 657986 153157 151178 364564 246004 067041 > 2139 [i]