Best Known (72−10, 72, s)-Nets in Base 4
(72−10, 72, 209717)-Net over F4 — Constructive and digital
Digital (62, 72, 209717)-net over F4, using
- 41 times duplication [i] based on digital (61, 71, 209717)-net over F4, using
- net defined by OOA [i] based on linear OOA(471, 209717, F4, 10, 10) (dual of [(209717, 10), 2097099, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(471, 1048585, F4, 10) (dual of [1048585, 1048514, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(40, 10, F4, 0) (dual of [10, 10, 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(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(471, 1048586, F4, 10) (dual of [1048586, 1048515, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(471, 1048585, F4, 10) (dual of [1048585, 1048514, 11]-code), using
- net defined by OOA [i] based on linear OOA(471, 209717, F4, 10, 10) (dual of [(209717, 10), 2097099, 11]-NRT-code), using
(72−10, 72, 524294)-Net over F4 — Digital
Digital (62, 72, 524294)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(472, 524294, F4, 2, 10) (dual of [(524294, 2), 1048516, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(472, 1048588, F4, 10) (dual of [1048588, 1048516, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(471, 1048576, F4, 10) (dual of [1048576, 1048505, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(461, 1048576, F4, 9) (dual of [1048576, 1048515, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(411, 12, F4, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,4)), using
- dual of repetition code with length 12 [i]
- linear OA(41, 12, F4, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(9) ⊂ Ce(8) [i] based on
- OOA 2-folding [i] based on linear OA(472, 1048588, F4, 10) (dual of [1048588, 1048516, 11]-code), using
(72−10, 72, large)-Net in Base 4 — Upper bound on s
There is no (62, 72, large)-net in base 4, because
- 8 times m-reduction [i] would yield (62, 64, large)-net in base 4, but