Best Known (72−11, 72, s)-Nets in Base 4
(72−11, 72, 13124)-Net over F4 — Constructive and digital
Digital (61, 72, 13124)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 7, 16)-net over F4, using
- digital (54, 65, 13108)-net over F4, using
- net defined by OOA [i] based on linear OOA(465, 13108, F4, 11, 11) (dual of [(13108, 11), 144123, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(465, 65541, F4, 11) (dual of [65541, 65476, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(465, 65544, F4, 11) (dual of [65544, 65479, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(465, 65536, F4, 11) (dual of [65536, 65471, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(465, 65544, F4, 11) (dual of [65544, 65479, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(465, 65541, F4, 11) (dual of [65541, 65476, 12]-code), using
- net defined by OOA [i] based on linear OOA(465, 13108, F4, 11, 11) (dual of [(13108, 11), 144123, 12]-NRT-code), using
(72−11, 72, 65575)-Net over F4 — Digital
Digital (61, 72, 65575)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(472, 65575, F4, 11) (dual of [65575, 65503, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
- linear OA(465, 65536, F4, 11) (dual of [65536, 65471, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(433, 65536, F4, 6) (dual of [65536, 65503, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(10) ⊂ Ce(5) [i] based on
(72−11, 72, large)-Net in Base 4 — Upper bound on s
There is no (61, 72, large)-net in base 4, because
- 9 times m-reduction [i] would yield (61, 63, large)-net in base 4, but