Best Known (31, 31+6, s)-Nets in Base 4
(31, 31+6, 87384)-Net over F4 — Constructive and digital
Digital (31, 37, 87384)-net over F4, using
- net defined by OOA [i] based on linear OOA(437, 87384, F4, 6, 6) (dual of [(87384, 6), 524267, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(437, 262152, F4, 6) (dual of [262152, 262115, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(437, 262153, F4, 6) (dual of [262153, 262116, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(428, 262144, F4, 5) (dual of [262144, 262116, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(437, 262153, F4, 6) (dual of [262153, 262116, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(437, 262152, F4, 6) (dual of [262152, 262115, 7]-code), using
(31, 31+6, 193404)-Net over F4 — Digital
Digital (31, 37, 193404)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(437, 193404, F4, 6) (dual of [193404, 193367, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(437, 262144, F4, 6) (dual of [262144, 262107, 7]-code), using
(31, 31+6, large)-Net in Base 4 — Upper bound on s
There is no (31, 37, large)-net in base 4, because
- 4 times m-reduction [i] would yield (31, 33, large)-net in base 4, but