Best Known (75−11, 75, s)-Nets in Base 2
(75−11, 75, 3280)-Net over F2 — Constructive and digital
Digital (64, 75, 3280)-net over F2, using
- 22 times duplication [i] based on digital (62, 73, 3280)-net over F2, using
- net defined by OOA [i] based on linear OOA(273, 3280, F2, 11, 11) (dual of [(3280, 11), 36007, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(273, 16401, F2, 11) (dual of [16401, 16328, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(272, 16400, F2, 11) (dual of [16400, 16328, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(215, 16, F2, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,2)), using
- dual of repetition code with length 16 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(272, 16400, F2, 11) (dual of [16400, 16328, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(273, 16401, F2, 11) (dual of [16401, 16328, 12]-code), using
- net defined by OOA [i] based on linear OOA(273, 3280, F2, 11, 11) (dual of [(3280, 11), 36007, 12]-NRT-code), using
(75−11, 75, 4210)-Net over F2 — Digital
Digital (64, 75, 4210)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(275, 4210, F2, 3, 11) (dual of [(4210, 3), 12555, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(275, 5467, F2, 3, 11) (dual of [(5467, 3), 16326, 12]-NRT-code), using
- 22 times duplication [i] based on linear OOA(273, 5467, F2, 3, 11) (dual of [(5467, 3), 16328, 12]-NRT-code), using
- OOA 3-folding [i] based on linear OA(273, 16401, F2, 11) (dual of [16401, 16328, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(272, 16400, F2, 11) (dual of [16400, 16328, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(257, 16384, F2, 9) (dual of [16384, 16327, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(215, 16, F2, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,2)), using
- dual of repetition code with length 16 [i]
- linear OA(21, 16, F2, 1) (dual of [16, 15, 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(10) ⊂ Ce(8) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(272, 16400, F2, 11) (dual of [16400, 16328, 12]-code), using
- OOA 3-folding [i] based on linear OA(273, 16401, F2, 11) (dual of [16401, 16328, 12]-code), using
- 22 times duplication [i] based on linear OOA(273, 5467, F2, 3, 11) (dual of [(5467, 3), 16328, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(275, 5467, F2, 3, 11) (dual of [(5467, 3), 16326, 12]-NRT-code), using
(75−11, 75, 74308)-Net in Base 2 — Upper bound on s
There is no (64, 75, 74309)-net in base 2, because
- 1 times m-reduction [i] would yield (64, 74, 74309)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 18889 931192 332237 303818 > 274 [i]