Best Known (212−30, 212, s)-Nets in Base 4
(212−30, 212, 17480)-Net over F4 — Constructive and digital
Digital (182, 212, 17480)-net over F4, using
- net defined by OOA [i] based on linear OOA(4212, 17480, F4, 30, 30) (dual of [(17480, 30), 524188, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4212, 262200, F4, 30) (dual of [262200, 261988, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4212, 262202, F4, 30) (dual of [262202, 261990, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(413, 58, F4, 6) (dual of [58, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(4212, 262202, F4, 30) (dual of [262202, 261990, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4212, 262200, F4, 30) (dual of [262200, 261988, 31]-code), using
(212−30, 212, 131101)-Net over F4 — Digital
Digital (182, 212, 131101)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4212, 131101, F4, 2, 30) (dual of [(131101, 2), 261990, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4212, 262202, F4, 30) (dual of [262202, 261990, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- linear OA(4199, 262144, F4, 30) (dual of [262144, 261945, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4154, 262144, F4, 23) (dual of [262144, 261990, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(413, 58, F4, 6) (dual of [58, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(4212, 262202, F4, 30) (dual of [262202, 261990, 31]-code), using
(212−30, 212, large)-Net in Base 4 — Upper bound on s
There is no (182, 212, large)-net in base 4, because
- 28 times m-reduction [i] would yield (182, 184, large)-net in base 4, but