Best Known (72−10, 72, s)-Nets in Base 2
(72−10, 72, 3280)-Net over F2 — Constructive and digital
Digital (62, 72, 3280)-net over F2, using
- net defined by OOA [i] based on linear OOA(272, 3280, F2, 10, 10) (dual of [(3280, 10), 32728, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(272, 16400, F2, 10) (dual of [16400, 16328, 11]-code), using
- strength reduction [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
- strength reduction [i] based on linear OA(272, 16400, F2, 11) (dual of [16400, 16328, 12]-code), using
- OA 5-folding and stacking [i] based on linear OA(272, 16400, F2, 10) (dual of [16400, 16328, 11]-code), using
(72−10, 72, 5466)-Net over F2 — Digital
Digital (62, 72, 5466)-net over F2, using
- 21 times duplication [i] based on digital (61, 71, 5466)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 5466, F2, 3, 10) (dual of [(5466, 3), 16327, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(271, 16398, F2, 10) (dual of [16398, 16327, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(271, 16399, F2, 10) (dual of [16399, 16328, 11]-code), using
- 1 times truncation [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 truncation [i] based on linear OA(272, 16400, F2, 11) (dual of [16400, 16328, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(271, 16399, F2, 10) (dual of [16399, 16328, 11]-code), using
- OOA 3-folding [i] based on linear OA(271, 16398, F2, 10) (dual of [16398, 16327, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(271, 5466, F2, 3, 10) (dual of [(5466, 3), 16327, 11]-NRT-code), using
(72−10, 72, 56313)-Net in Base 2 — Upper bound on s
There is no (62, 72, 56314)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 4722 484333 746803 568374 > 272 [i]