Best Known (134, 134+14, s)-Nets in Base 2
(134, 134+14, 299596)-Net over F2 — Constructive and digital
Digital (134, 148, 299596)-net over F2, using
- net defined by OOA [i] based on linear OOA(2148, 299596, F2, 14, 14) (dual of [(299596, 14), 4194196, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2148, 2097172, F2, 14) (dual of [2097172, 2097024, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2148, 2097174, F2, 14) (dual of [2097174, 2097026, 15]-code), using
- 1 times truncation [i] based on linear OA(2149, 2097175, F2, 15) (dual of [2097175, 2097026, 16]-code), using
- construction X4 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(222, 23, F2, 21) (dual of [23, 1, 22]-code), using
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- dual of repetition code with length 23 [i]
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- 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 X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(2149, 2097175, F2, 15) (dual of [2097175, 2097026, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2148, 2097174, F2, 14) (dual of [2097174, 2097026, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2148, 2097172, F2, 14) (dual of [2097172, 2097024, 15]-code), using
(134, 134+14, 419434)-Net over F2 — Digital
Digital (134, 148, 419434)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2148, 419434, F2, 5, 14) (dual of [(419434, 5), 2097022, 15]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2148, 2097170, F2, 14) (dual of [2097170, 2097022, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2148, 2097174, F2, 14) (dual of [2097174, 2097026, 15]-code), using
- 1 times truncation [i] based on linear OA(2149, 2097175, F2, 15) (dual of [2097175, 2097026, 16]-code), using
- construction X4 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(222, 23, F2, 21) (dual of [23, 1, 22]-code), using
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- dual of repetition code with length 23 [i]
- strength reduction [i] based on linear OA(222, 23, F2, 22) (dual of [23, 1, 23]-code or 23-arc in PG(21,2)), using
- 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 X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(2149, 2097175, F2, 15) (dual of [2097175, 2097026, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2148, 2097174, F2, 14) (dual of [2097174, 2097026, 15]-code), using
- OOA 5-folding [i] based on linear OA(2148, 2097170, F2, 14) (dual of [2097170, 2097022, 15]-code), using
(134, 134+14, 7826224)-Net in Base 2 — Upper bound on s
There is no (134, 148, 7826225)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 356 812166 504191 469446 591687 418554 780827 856896 > 2148 [i]