Best Known (62−13, 62, s)-Nets in Base 2
(62−13, 62, 172)-Net over F2 — Constructive and digital
Digital (49, 62, 172)-net over F2, using
- net defined by OOA [i] based on linear OOA(262, 172, F2, 13, 13) (dual of [(172, 13), 2174, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(262, 1033, F2, 13) (dual of [1033, 971, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(262, 1035, F2, 13) (dual of [1035, 973, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(261, 1024, F2, 13) (dual of [1024, 963, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(251, 1024, F2, 11) (dual of [1024, 973, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(262, 1035, F2, 13) (dual of [1035, 973, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(262, 1033, F2, 13) (dual of [1033, 971, 14]-code), using
(62−13, 62, 345)-Net over F2 — Digital
Digital (49, 62, 345)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(262, 345, F2, 3, 13) (dual of [(345, 3), 973, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(262, 1035, F2, 13) (dual of [1035, 973, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(261, 1024, F2, 13) (dual of [1024, 963, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(251, 1024, F2, 11) (dual of [1024, 973, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(21, 11, F2, 1) (dual of [11, 10, 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(12) ⊂ Ce(10) [i] based on
- OOA 3-folding [i] based on linear OA(262, 1035, F2, 13) (dual of [1035, 973, 14]-code), using
(62−13, 62, 3432)-Net in Base 2 — Upper bound on s
There is no (49, 62, 3433)-net in base 2, because
- 1 times m-reduction [i] would yield (49, 61, 3433)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2 307529 348957 025972 > 261 [i]