Best Known (134, 134+15, s)-Nets in Base 2
(134, 134+15, 299596)-Net over F2 — Constructive and digital
Digital (134, 149, 299596)-net over F2, using
- net defined by OOA [i] based on linear OOA(2149, 299596, F2, 15, 15) (dual of [(299596, 15), 4493791, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2149, 2097173, F2, 15) (dual of [2097173, 2097024, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2149, 2097174, F2, 15) (dual of [2097174, 2097025, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2148, 2097152, F2, 15) (dual of [2097152, 2097004, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2127, 2097152, F2, 13) (dual of [2097152, 2097025, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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 X applied to Ce(14) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(2149, 2097174, F2, 15) (dual of [2097174, 2097025, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2149, 2097173, F2, 15) (dual of [2097173, 2097024, 16]-code), using
(134, 134+15, 349529)-Net over F2 — Digital
Digital (134, 149, 349529)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2149, 349529, F2, 6, 15) (dual of [(349529, 6), 2097025, 16]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2149, 2097174, F2, 15) (dual of [2097174, 2097025, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2148, 2097152, F2, 15) (dual of [2097152, 2097004, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2127, 2097152, F2, 13) (dual of [2097152, 2097025, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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 X applied to Ce(14) ⊂ Ce(12) [i] based on
- OOA 6-folding [i] based on linear OA(2149, 2097174, F2, 15) (dual of [2097174, 2097025, 16]-code), using
(134, 134+15, 7826224)-Net in Base 2 — Upper bound on s
There is no (134, 149, 7826225)-net in base 2, because
- 1 times m-reduction [i] would yield (134, 148, 7826225)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 356 812166 504191 469446 591687 418554 780827 856896 > 2148 [i]