Best Known (81, 92, s)-Nets in Base 4
(81, 92, 838866)-Net over F4 — Constructive and digital
Digital (81, 92, 838866)-net over F4, using
- net defined by OOA [i] based on linear OOA(492, 838866, F4, 11, 11) (dual of [(838866, 11), 9227434, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(492, 4194331, F4, 11) (dual of [4194331, 4194239, 12]-code), using
- 2 times code embedding in larger space [i] based on linear OA(490, 4194329, F4, 11) (dual of [4194329, 4194239, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(489, 4194305, F4, 11) (dual of [4194305, 4194216, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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([0,5]) ⊂ C([0,4]) [i] based on
- 2 times code embedding in larger space [i] based on linear OA(490, 4194329, F4, 11) (dual of [4194329, 4194239, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(492, 4194331, F4, 11) (dual of [4194331, 4194239, 12]-code), using
(81, 92, 2097166)-Net over F4 — Digital
Digital (81, 92, 2097166)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(492, 2097166, F4, 2, 11) (dual of [(2097166, 2), 4194240, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(492, 4194332, F4, 11) (dual of [4194332, 4194240, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(490, 4194329, F4, 11) (dual of [4194329, 4194239, 12]-code), using
- construction X4 applied to C([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(489, 4194305, F4, 11) (dual of [4194305, 4194216, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(467, 4194305, F4, 9) (dual of [4194305, 4194238, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4194305 | 422−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(423, 24, F4, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,4)), using
- dual of repetition code with length 24 [i]
- linear OA(41, 24, F4, 1) (dual of [24, 23, 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([0,5]) ⊂ C([0,4]) [i] based on
- linear OA(490, 4194330, F4, 9) (dual of [4194330, 4194240, 10]-code), using Gilbert–Varšamov bound and bm = 490 > Vbs−1(k−1) = 15586 436580 914441 471956 621644 321929 588584 783055 727931 [i]
- linear OA(41, 2, F4, 1) (dual of [2, 1, 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 (see above)
- linear OA(490, 4194329, F4, 11) (dual of [4194329, 4194239, 12]-code), using
- construction X with Varšamov bound [i] based on
- OOA 2-folding [i] based on linear OA(492, 4194332, F4, 11) (dual of [4194332, 4194240, 12]-code), using
(81, 92, large)-Net in Base 4 — Upper bound on s
There is no (81, 92, large)-net in base 4, because
- 9 times m-reduction [i] would yield (81, 83, large)-net in base 4, but