Best Known (59−10, 59, s)-Nets in Base 4
(59−10, 59, 13109)-Net over F4 — Constructive and digital
Digital (49, 59, 13109)-net over F4, using
- 41 times duplication [i] based on digital (48, 58, 13109)-net over F4, using
- net defined by OOA [i] based on linear OOA(458, 13109, F4, 10, 10) (dual of [(13109, 10), 131032, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(458, 65545, F4, 10) (dual of [65545, 65487, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(458, 65546, F4, 10) (dual of [65546, 65488, 11]-code), using
- construction X4 applied to C([0,9]) ⊂ C([1,8]) [i] based on
- linear OA(457, 65535, F4, 10) (dual of [65535, 65478, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(448, 65535, F4, 8) (dual of [65535, 65487, 9]-code), using 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(410, 11, F4, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,4)), using
- dual of repetition code with length 11 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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([0,9]) ⊂ C([1,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(458, 65546, F4, 10) (dual of [65546, 65488, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(458, 65545, F4, 10) (dual of [65545, 65487, 11]-code), using
- net defined by OOA [i] based on linear OOA(458, 13109, F4, 10, 10) (dual of [(13109, 10), 131032, 11]-NRT-code), using
(59−10, 59, 32773)-Net over F4 — Digital
Digital (49, 59, 32773)-net over F4, using
- 41 times duplication [i] based on digital (48, 58, 32773)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(458, 32773, F4, 2, 10) (dual of [(32773, 2), 65488, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(458, 65546, F4, 10) (dual of [65546, 65488, 11]-code), using
- construction X4 applied to C([0,9]) ⊂ C([1,8]) [i] based on
- linear OA(457, 65535, F4, 10) (dual of [65535, 65478, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(448, 65535, F4, 8) (dual of [65535, 65487, 9]-code), using 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(410, 11, F4, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,4)), using
- dual of repetition code with length 11 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 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([0,9]) ⊂ C([1,8]) [i] based on
- OOA 2-folding [i] based on linear OA(458, 65546, F4, 10) (dual of [65546, 65488, 11]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(458, 32773, F4, 2, 10) (dual of [(32773, 2), 65488, 11]-NRT-code), using
(59−10, 59, large)-Net in Base 4 — Upper bound on s
There is no (49, 59, large)-net in base 4, because
- 8 times m-reduction [i] would yield (49, 51, large)-net in base 4, but