Best Known (73, 73+14, s)-Nets in Base 4
(73, 73+14, 9366)-Net over F4 — Constructive and digital
Digital (73, 87, 9366)-net over F4, using
- 41 times duplication [i] based on digital (72, 86, 9366)-net over F4, using
- net defined by OOA [i] based on linear OOA(486, 9366, F4, 14, 14) (dual of [(9366, 14), 131038, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(486, 65562, F4, 14) (dual of [65562, 65476, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(486, 65565, F4, 14) (dual of [65565, 65479, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(13) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(486, 65565, F4, 14) (dual of [65565, 65479, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(486, 65562, F4, 14) (dual of [65562, 65476, 15]-code), using
- net defined by OOA [i] based on linear OOA(486, 9366, F4, 14, 14) (dual of [(9366, 14), 131038, 15]-NRT-code), using
(73, 73+14, 36383)-Net over F4 — Digital
Digital (73, 87, 36383)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(487, 36383, F4, 14) (dual of [36383, 36296, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(487, 65567, F4, 14) (dual of [65567, 65480, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(457, 65536, F4, 10) (dual of [65536, 65479, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(449, 65536, F4, 9) (dual of [65536, 65487, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(45, 30, F4, 3) (dual of [30, 25, 4]-code or 30-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(13) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(487, 65567, F4, 14) (dual of [65567, 65480, 15]-code), using
(73, 73+14, large)-Net in Base 4 — Upper bound on s
There is no (73, 87, large)-net in base 4, because
- 12 times m-reduction [i] would yield (73, 75, large)-net in base 4, but