Best Known (212, 212+23, s)-Nets in Base 2
(212, 212+23, 190652)-Net over F2 — Constructive and digital
Digital (212, 235, 190652)-net over F2, using
- 22 times duplication [i] based on digital (210, 233, 190652)-net over F2, using
- net defined by OOA [i] based on linear OOA(2233, 190652, F2, 23, 23) (dual of [(190652, 23), 4384763, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2233, 2097173, F2, 23) (dual of [2097173, 2096940, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(2233, 2097174, F2, 23) (dual of [2097174, 2096941, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(20) [i] based on
- linear OA(2232, 2097152, F2, 23) (dual of [2097152, 2096920, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(2211, 2097152, F2, 21) (dual of [2097152, 2096941, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(22) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2233, 2097174, F2, 23) (dual of [2097174, 2096941, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(2233, 2097173, F2, 23) (dual of [2097173, 2096940, 24]-code), using
- net defined by OOA [i] based on linear OOA(2233, 190652, F2, 23, 23) (dual of [(190652, 23), 4384763, 24]-NRT-code), using
(212, 212+23, 262147)-Net over F2 — Digital
Digital (212, 235, 262147)-net over F2, using
- 21 times duplication [i] based on digital (211, 234, 262147)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2234, 262147, F2, 8, 23) (dual of [(262147, 8), 2096942, 24]-NRT-code), using
- OOA 8-folding [i] based on linear OA(2234, 2097176, F2, 23) (dual of [2097176, 2096942, 24]-code), using
- adding a parity check bit [i] based on linear OA(2233, 2097175, F2, 22) (dual of [2097175, 2096942, 23]-code), using
- construction X4 applied to C([0,22]) ⊂ C([1,20]) [i] based on
- linear OA(2232, 2097151, F2, 23) (dual of [2097151, 2096919, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 2097151 = 221−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2210, 2097151, F2, 20) (dual of [2097151, 2096941, 21]-code), using the primitive narrow-sense BCH-code C(I) with length 2097151 = 221−1, 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 C([0,22]) ⊂ C([1,20]) [i] based on
- adding a parity check bit [i] based on linear OA(2233, 2097175, F2, 22) (dual of [2097175, 2096942, 23]-code), using
- OOA 8-folding [i] based on linear OA(2234, 2097176, F2, 23) (dual of [2097176, 2096942, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2234, 262147, F2, 8, 23) (dual of [(262147, 8), 2096942, 24]-NRT-code), using
(212, 212+23, large)-Net in Base 2 — Upper bound on s
There is no (212, 235, large)-net in base 2, because
- 21 times m-reduction [i] would yield (212, 214, large)-net in base 2, but