Best Known (118−13, 118, s)-Nets in Base 2
(118−13, 118, 87385)-Net over F2 — Constructive and digital
Digital (105, 118, 87385)-net over F2, using
- net defined by OOA [i] based on linear OOA(2118, 87385, F2, 13, 13) (dual of [(87385, 13), 1135887, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2118, 524311, F2, 13) (dual of [524311, 524193, 14]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2116, 524309, F2, 13) (dual of [524309, 524193, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2115, 524288, F2, 13) (dual of [524288, 524173, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(296, 524288, F2, 11) (dual of [524288, 524192, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(220, 21, F2, 19) (dual of [21, 1, 20]-code), using
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- dual of repetition code with length 21 [i]
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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(12) ⊂ Ce(10) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2116, 524309, F2, 13) (dual of [524309, 524193, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2118, 524311, F2, 13) (dual of [524311, 524193, 14]-code), using
(118−13, 118, 104862)-Net over F2 — Digital
Digital (105, 118, 104862)-net over F2, using
- 21 times duplication [i] based on digital (104, 117, 104862)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2117, 104862, F2, 5, 13) (dual of [(104862, 5), 524193, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2117, 524310, F2, 13) (dual of [524310, 524193, 14]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2116, 524309, F2, 13) (dual of [524309, 524193, 14]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(2115, 524288, F2, 13) (dual of [524288, 524173, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(296, 524288, F2, 11) (dual of [524288, 524192, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(220, 21, F2, 19) (dual of [21, 1, 20]-code), using
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- dual of repetition code with length 21 [i]
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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(12) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2116, 524309, F2, 13) (dual of [524309, 524193, 14]-code), using
- OOA 5-folding [i] based on linear OA(2117, 524310, F2, 13) (dual of [524310, 524193, 14]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2117, 104862, F2, 5, 13) (dual of [(104862, 5), 524193, 14]-NRT-code), using
(118−13, 118, 2219756)-Net in Base 2 — Upper bound on s
There is no (105, 118, 2219757)-net in base 2, because
- 1 times m-reduction [i] would yield (105, 117, 2219757)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 166153 726815 974663 011631 653718 388464 > 2117 [i]