Best Known (123, 123+15, s)-Nets in Base 2
(123, 123+15, 74901)-Net over F2 — Constructive and digital
Digital (123, 138, 74901)-net over F2, using
- 23 times duplication [i] based on digital (120, 135, 74901)-net over F2, using
- net defined by OOA [i] based on linear OOA(2135, 74901, F2, 15, 15) (dual of [(74901, 15), 1123380, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2135, 524308, F2, 15) (dual of [524308, 524173, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(21, 20, F2, 1) (dual of [20, 19, 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 7-folding and stacking with additional row [i] based on linear OA(2135, 524308, F2, 15) (dual of [524308, 524173, 16]-code), using
- net defined by OOA [i] based on linear OOA(2135, 74901, F2, 15, 15) (dual of [(74901, 15), 1123380, 16]-NRT-code), using
(123, 123+15, 104862)-Net over F2 — Digital
Digital (123, 138, 104862)-net over F2, using
- 21 times duplication [i] based on digital (122, 137, 104862)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 104862, F2, 5, 15) (dual of [(104862, 5), 524173, 16]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2136, 104862, F2, 5, 15) (dual of [(104862, 5), 524174, 16]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2136, 524310, F2, 15) (dual of [524310, 524174, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2135, 524309, F2, 15) (dual of [524309, 524174, 16]-code), using
- construction X4 applied to Ce(14) ⊂ Ce(12) [i] based on
- linear OA(2134, 524288, F2, 15) (dual of [524288, 524154, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 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(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(14) ⊂ Ce(12) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2135, 524309, F2, 15) (dual of [524309, 524174, 16]-code), using
- OOA 5-folding [i] based on linear OA(2136, 524310, F2, 15) (dual of [524310, 524174, 16]-code), using
- 21 times duplication [i] based on linear OOA(2136, 104862, F2, 5, 15) (dual of [(104862, 5), 524174, 16]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2137, 104862, F2, 5, 15) (dual of [(104862, 5), 524173, 16]-NRT-code), using
(123, 123+15, 2633322)-Net in Base 2 — Upper bound on s
There is no (123, 138, 2633323)-net in base 2, because
- 1 times m-reduction [i] would yield (123, 137, 2633323)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 174224 783979 563740 637807 849916 685034 315428 > 2137 [i]