Best Known (213−27, 213, s)-Nets in Base 4
(213−27, 213, 80664)-Net over F4 — Constructive and digital
Digital (186, 213, 80664)-net over F4, using
- 42 times duplication [i] based on digital (184, 211, 80664)-net over F4, using
- net defined by OOA [i] based on linear OOA(4211, 80664, F4, 27, 27) (dual of [(80664, 27), 2177717, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4211, 1048633, F4, 27) (dual of [1048633, 1048422, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(4211, 1048636, F4, 27) (dual of [1048636, 1048425, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4211, 1048636, F4, 27) (dual of [1048636, 1048425, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(4211, 1048633, F4, 27) (dual of [1048633, 1048422, 28]-code), using
- net defined by OOA [i] based on linear OOA(4211, 80664, F4, 27, 27) (dual of [(80664, 27), 2177717, 28]-NRT-code), using
(213−27, 213, 524319)-Net over F4 — Digital
Digital (186, 213, 524319)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4213, 524319, F4, 2, 27) (dual of [(524319, 2), 1048425, 28]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4211, 524318, F4, 2, 27) (dual of [(524318, 2), 1048425, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4211, 1048636, F4, 27) (dual of [1048636, 1048425, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- linear OA(4201, 1048576, F4, 27) (dual of [1048576, 1048375, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(4151, 1048576, F4, 21) (dual of [1048576, 1048425, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to Ce(26) ⊂ Ce(20) [i] based on
- OOA 2-folding [i] based on linear OA(4211, 1048636, F4, 27) (dual of [1048636, 1048425, 28]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4211, 524318, F4, 2, 27) (dual of [(524318, 2), 1048425, 28]-NRT-code), using
(213−27, 213, large)-Net in Base 4 — Upper bound on s
There is no (186, 213, large)-net in base 4, because
- 25 times m-reduction [i] would yield (186, 188, large)-net in base 4, but