Best Known (100−12, 100, s)-Nets in Base 4
(100−12, 100, 699052)-Net over F4 — Constructive and digital
Digital (88, 100, 699052)-net over F4, using
- net defined by OOA [i] based on linear OOA(4100, 699052, F4, 12, 12) (dual of [(699052, 12), 8388524, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(4100, 4194312, F4, 12) (dual of [4194312, 4194212, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(4100, 4194316, F4, 12) (dual of [4194316, 4194216, 13]-code), using
- construction X4 applied to C([1,12]) ⊂ C([1,10]) [i] based on
- linear OA(499, 4194303, F4, 12) (dual of [4194303, 4194204, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(488, 4194303, F4, 10) (dual of [4194303, 4194215, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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 C([1,12]) ⊂ C([1,10]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4100, 4194316, F4, 12) (dual of [4194316, 4194216, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(4100, 4194312, F4, 12) (dual of [4194312, 4194212, 13]-code), using
(100−12, 100, 2097158)-Net over F4 — Digital
Digital (88, 100, 2097158)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4100, 2097158, F4, 2, 12) (dual of [(2097158, 2), 4194216, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4100, 4194316, F4, 12) (dual of [4194316, 4194216, 13]-code), using
- construction X4 applied to C([1,12]) ⊂ C([1,10]) [i] based on
- linear OA(499, 4194303, F4, 12) (dual of [4194303, 4194204, 13]-code), using the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(488, 4194303, F4, 10) (dual of [4194303, 4194215, 11]-code), using the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(412, 13, F4, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,4)), using
- dual of repetition code with length 13 [i]
- linear OA(41, 13, F4, 1) (dual of [13, 12, 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 C([1,12]) ⊂ C([1,10]) [i] based on
- OOA 2-folding [i] based on linear OA(4100, 4194316, F4, 12) (dual of [4194316, 4194216, 13]-code), using
(100−12, 100, large)-Net in Base 4 — Upper bound on s
There is no (88, 100, large)-net in base 4, because
- 10 times m-reduction [i] would yield (88, 90, large)-net in base 4, but