Best Known (72, 72+21, s)-Nets in Base 3
(72, 72+21, 464)-Net over F3 — Constructive and digital
Digital (72, 93, 464)-net over F3, using
- 31 times duplication [i] based on digital (71, 92, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 23, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 23, 116)-net over F81, using
(72, 72+21, 778)-Net over F3 — Digital
Digital (72, 93, 778)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(393, 778, F3, 21) (dual of [778, 685, 22]-code), using
- 30 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0) [i] based on linear OA(385, 740, F3, 21) (dual of [740, 655, 22]-code), using
- construction XX applied to C1 = C([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,18}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(385, 728, F3, 21) (dual of [728, 643, 22]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,19}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(373, 728, F3, 19) (dual of [728, 655, 20]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,18], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code) (see above)
- construction XX applied to C1 = C([727,18]), C2 = C([0,19]), C3 = C1 + C2 = C([0,18]), and C∩ = C1 ∩ C2 = C([727,19]) [i] based on
- 30 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0) [i] based on linear OA(385, 740, F3, 21) (dual of [740, 655, 22]-code), using
(72, 72+21, 55512)-Net in Base 3 — Upper bound on s
There is no (72, 93, 55513)-net in base 3, because
- 1 times m-reduction [i] would yield (72, 92, 55513)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 78 564617 085722 053617 850078 234540 879320 449817 > 392 [i]