Best Known (38−7, 38, s)-Nets in Base 4
(38−7, 38, 5466)-Net over F4 — Constructive and digital
Digital (31, 38, 5466)-net over F4, using
- 41 times duplication [i] based on digital (30, 37, 5466)-net over F4, using
- net defined by OOA [i] based on linear OOA(437, 5466, F4, 7, 7) (dual of [(5466, 7), 38225, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(437, 16399, F4, 7) (dual of [16399, 16362, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(41, 15, F4, 1) (dual of [15, 14, 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 X applied to Ce(6) ⊂ Ce(4) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(437, 16399, F4, 7) (dual of [16399, 16362, 8]-code), using
- net defined by OOA [i] based on linear OOA(437, 5466, F4, 7, 7) (dual of [(5466, 7), 38225, 8]-NRT-code), using
(38−7, 38, 16402)-Net over F4 — Digital
Digital (31, 38, 16402)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(438, 16402, F4, 7) (dual of [16402, 16364, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(437, 16400, F4, 7) (dual of [16400, 16363, 8]-code), using
- construction X4 applied to Ce(6) ⊂ Ce(4) [i] based on
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(415, 16, F4, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,4)), using
- dual of repetition code with length 16 [i]
- linear OA(41, 16, F4, 1) (dual of [16, 15, 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 Ce(6) ⊂ Ce(4) [i] based on
- linear OA(437, 16401, F4, 6) (dual of [16401, 16364, 7]-code), using Gilbert–Varšamov bound and bm = 437 > Vbs−1(k−1) = 2401 173698 317620 887941 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 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
- linear OA(437, 16400, F4, 7) (dual of [16400, 16363, 8]-code), using
- construction X with Varšamov bound [i] based on
(38−7, 38, large)-Net in Base 4 — Upper bound on s
There is no (31, 38, large)-net in base 4, because
- 5 times m-reduction [i] would yield (31, 33, large)-net in base 4, but