Best Known (245−23, 245, s)-Nets in Base 2
(245−23, 245, 381302)-Net over F2 — Constructive and digital
Digital (222, 245, 381302)-net over F2, using
- 21 times duplication [i] based on digital (221, 244, 381302)-net over F2, using
- net defined by OOA [i] based on linear OOA(2244, 381302, F2, 23, 23) (dual of [(381302, 23), 8769702, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2244, 4194323, F2, 23) (dual of [4194323, 4194079, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2244, 4194327, F2, 23) (dual of [4194327, 4194083, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2243, 4194304, F2, 23) (dual of [4194304, 4194061, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2221, 4194304, F2, 21) (dual of [4194304, 4194083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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 X applied to Ce(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2244, 4194327, F2, 23) (dual of [4194327, 4194083, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2244, 4194323, F2, 23) (dual of [4194323, 4194079, 24]-code), using
- net defined by OOA [i] based on linear OOA(2244, 381302, F2, 23, 23) (dual of [(381302, 23), 8769702, 24]-NRT-code), using
(245−23, 245, 524291)-Net over F2 — Digital
Digital (222, 245, 524291)-net over F2, using
- 21 times duplication [i] based on digital (221, 244, 524291)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 524291, F2, 8, 23) (dual of [(524291, 8), 4194084, 24]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2244, 4194328, F2, 23) (dual of [4194328, 4194084, 24]-code), using
- construction X4 applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2243, 4194304, F2, 23) (dual of [4194304, 4194061, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2221, 4194304, F2, 21) (dual of [4194304, 4194083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 222−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(223, 24, F2, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,2)), using
- dual of repetition code with length 24 [i]
- linear OA(21, 24, F2, 1) (dual of [24, 23, 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(22) ⊂ Ce(20) [i] based on
- OOA 8-folding [i] based on linear OA(2244, 4194328, F2, 23) (dual of [4194328, 4194084, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2244, 524291, F2, 8, 23) (dual of [(524291, 8), 4194084, 24]-NRT-code), using
(245−23, 245, large)-Net in Base 2 — Upper bound on s
There is no (222, 245, large)-net in base 2, because
- 21 times m-reduction [i] would yield (222, 224, large)-net in base 2, but