Best Known (72, 72+17, s)-Nets in Base 4
(72, 72+17, 2050)-Net over F4 — Constructive and digital
Digital (72, 89, 2050)-net over F4, using
- 41 times duplication [i] based on digital (71, 88, 2050)-net over F4, using
- net defined by OOA [i] based on linear OOA(488, 2050, F4, 17, 17) (dual of [(2050, 17), 34762, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(488, 16401, F4, 17) (dual of [16401, 16313, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(43, 17, F4, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(16) ⊂ Ce(13) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(488, 16401, F4, 17) (dual of [16401, 16313, 18]-code), using
- net defined by OOA [i] based on linear OOA(488, 2050, F4, 17, 17) (dual of [(2050, 17), 34762, 18]-NRT-code), using
(72, 72+17, 8201)-Net over F4 — Digital
Digital (72, 89, 8201)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(489, 8201, F4, 2, 17) (dual of [(8201, 2), 16313, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(489, 16402, F4, 17) (dual of [16402, 16313, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- linear OA(485, 16385, F4, 17) (dual of [16385, 16300, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(471, 16385, F4, 13) (dual of [16385, 16314, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 414−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- linear OA(44, 17, F4, 3) (dual of [17, 13, 4]-code or 17-cap in PG(3,4)), using
- construction X applied to C([0,8]) ⊂ C([0,6]) [i] based on
- OOA 2-folding [i] based on linear OA(489, 16402, F4, 17) (dual of [16402, 16313, 18]-code), using
(72, 72+17, 5262937)-Net in Base 4 — Upper bound on s
There is no (72, 89, 5262938)-net in base 4, because
- 1 times m-reduction [i] would yield (72, 88, 5262938)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 95781 054927 580927 445414 675538 207573 402612 602992 529695 > 488 [i]