Best Known (131, 131+15, s)-Nets in Base 2
(131, 131+15, 149800)-Net over F2 — Constructive and digital
Digital (131, 146, 149800)-net over F2, using
- 21 times duplication [i] based on digital (130, 145, 149800)-net over F2, using
- net defined by OOA [i] based on linear OOA(2145, 149800, F2, 15, 15) (dual of [(149800, 15), 2246855, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2145, 1048601, F2, 15) (dual of [1048601, 1048456, 16]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2121, 1048576, F2, 13) (dual of [1048576, 1048455, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(221, 22, F2, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,2)), using
- dual of repetition code with length 22 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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(14) ⊂ Ce(12) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2145, 1048601, F2, 15) (dual of [1048601, 1048456, 16]-code), using
- net defined by OOA [i] based on linear OOA(2145, 149800, F2, 15, 15) (dual of [(149800, 15), 2246855, 16]-NRT-code), using
(131, 131+15, 209720)-Net over F2 — Digital
Digital (131, 146, 209720)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2146, 209720, F2, 5, 15) (dual of [(209720, 5), 1048454, 16]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2144, 209720, F2, 5, 15) (dual of [(209720, 5), 1048456, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2144, 1048600, F2, 15) (dual of [1048600, 1048456, 16]-code), using
- 2 times code embedding in larger space [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2141, 1048576, F2, 15) (dual of [1048576, 1048435, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2121, 1048576, F2, 13) (dual of [1048576, 1048455, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(221, 22, F2, 21) (dual of [22, 1, 22]-code or 22-arc in PG(20,2)), using
- dual of repetition code with length 22 [i]
- linear OA(21, 22, F2, 1) (dual of [22, 21, 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(14) ⊂ Ce(12) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(2142, 1048598, F2, 15) (dual of [1048598, 1048456, 16]-code), using
- OOA 5-folding [i] based on linear OA(2144, 1048600, F2, 15) (dual of [1048600, 1048456, 16]-code), using
- 22 times duplication [i] based on linear OOA(2144, 209720, F2, 5, 15) (dual of [(209720, 5), 1048456, 16]-NRT-code), using
(131, 131+15, 5814859)-Net in Base 2 — Upper bound on s
There is no (131, 146, 5814860)-net in base 2, because
- 1 times m-reduction [i] would yield (131, 145, 5814860)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 44 601506 975302 700958 581610 894778 988806 538838 > 2145 [i]