Best Known (97, 97+19, s)-Nets in Base 4
(97, 97+19, 7283)-Net over F4 — Constructive and digital
Digital (97, 116, 7283)-net over F4, using
- 42 times duplication [i] based on digital (95, 114, 7283)-net over F4, using
- net defined by OOA [i] based on linear OOA(4114, 7283, F4, 19, 19) (dual of [(7283, 19), 138263, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4114, 65548, F4, 19) (dual of [65548, 65434, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(4114, 65554, F4, 19) (dual of [65554, 65440, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(4113, 65537, F4, 19) (dual of [65537, 65424, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4114, 65554, F4, 19) (dual of [65554, 65440, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(4114, 65548, F4, 19) (dual of [65548, 65434, 20]-code), using
- net defined by OOA [i] based on linear OOA(4114, 7283, F4, 19, 19) (dual of [(7283, 19), 138263, 20]-NRT-code), using
(97, 97+19, 32778)-Net over F4 — Digital
Digital (97, 116, 32778)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4116, 32778, F4, 2, 19) (dual of [(32778, 2), 65440, 20]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4114, 32777, F4, 2, 19) (dual of [(32777, 2), 65440, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4114, 65554, F4, 19) (dual of [65554, 65440, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,8]) [i] based on
- linear OA(4113, 65537, F4, 19) (dual of [65537, 65424, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(497, 65537, F4, 17) (dual of [65537, 65440, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(41, 17, F4, 1) (dual of [17, 16, 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 C([0,9]) ⊂ C([0,8]) [i] based on
- OOA 2-folding [i] based on linear OA(4114, 65554, F4, 19) (dual of [65554, 65440, 20]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4114, 32777, F4, 2, 19) (dual of [(32777, 2), 65440, 20]-NRT-code), using
(97, 97+19, large)-Net in Base 4 — Upper bound on s
There is no (97, 116, large)-net in base 4, because
- 17 times m-reduction [i] would yield (97, 99, large)-net in base 4, but