Best Known (98−18, 98, s)-Nets in Base 4
(98−18, 98, 1823)-Net over F4 — Constructive and digital
Digital (80, 98, 1823)-net over F4, using
- 41 times duplication [i] based on digital (79, 97, 1823)-net over F4, using
- net defined by OOA [i] based on linear OOA(497, 1823, F4, 18, 18) (dual of [(1823, 18), 32717, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(497, 16407, F4, 18) (dual of [16407, 16310, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(497, 16410, F4, 18) (dual of [16410, 16313, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-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(17) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(497, 16410, F4, 18) (dual of [16410, 16313, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(497, 16407, F4, 18) (dual of [16407, 16310, 19]-code), using
- net defined by OOA [i] based on linear OOA(497, 1823, F4, 18, 18) (dual of [(1823, 18), 32717, 19]-NRT-code), using
(98−18, 98, 10113)-Net over F4 — Digital
Digital (80, 98, 10113)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(498, 10113, F4, 18) (dual of [10113, 10015, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(498, 16412, F4, 18) (dual of [16412, 16314, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(45, 27, F4, 3) (dual of [27, 22, 4]-code or 27-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(17) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(498, 16412, F4, 18) (dual of [16412, 16314, 19]-code), using
(98−18, 98, 4970429)-Net in Base 4 — Upper bound on s
There is no (80, 98, 4970430)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 100433 690449 887941 710873 100481 013817 187623 821146 516921 514039 > 498 [i]