Best Known (112−10, 112, s)-Nets in Base 2
(112−10, 112, 838865)-Net over F2 — Constructive and digital
Digital (102, 112, 838865)-net over F2, using
- t-expansion [i] based on digital (101, 112, 838865)-net over F2, using
- net defined by OOA [i] based on linear OOA(2112, 838865, F2, 11, 11) (dual of [(838865, 11), 9227403, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2112, 4194326, F2, 11) (dual of [4194326, 4194214, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(2112, 4194327, F2, 11) (dual of [4194327, 4194215, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(2111, 4194304, F2, 11) (dual of [4194304, 4194193, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(289, 4194304, F2, 9) (dual of [4194304, 4194215, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(21, 23, F2, 1) (dual of [23, 22, 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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(2112, 4194327, F2, 11) (dual of [4194327, 4194215, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(2112, 4194326, F2, 11) (dual of [4194326, 4194214, 12]-code), using
- net defined by OOA [i] based on linear OOA(2112, 838865, F2, 11, 11) (dual of [(838865, 11), 9227403, 12]-NRT-code), using
(112−10, 112, 1048582)-Net over F2 — Digital
Digital (102, 112, 1048582)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2112, 1048582, F2, 4, 10) (dual of [(1048582, 4), 4194216, 11]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2112, 4194328, F2, 10) (dual of [4194328, 4194216, 11]-code), using
- strength reduction [i] based on linear OA(2112, 4194328, F2, 11) (dual of [4194328, 4194216, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(2111, 4194304, F2, 11) (dual of [4194304, 4194193, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(289, 4194304, F2, 9) (dual of [4194304, 4194215, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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(10) ⊂ Ce(8) [i] based on
- strength reduction [i] based on linear OA(2112, 4194328, F2, 11) (dual of [4194328, 4194216, 12]-code), using
- OOA 4-folding [i] based on linear OA(2112, 4194328, F2, 10) (dual of [4194328, 4194216, 11]-code), using
(112−10, 112, large)-Net in Base 2 — Upper bound on s
There is no (102, 112, large)-net in base 2, because
- 8 times m-reduction [i] would yield (102, 104, large)-net in base 2, but