Best Known (64, 79, s)-Nets in Base 5
(64, 79, 2237)-Net over F5 — Constructive and digital
Digital (64, 79, 2237)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 7, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (57, 72, 2231)-net over F5, using
- net defined by OOA [i] based on linear OOA(572, 2231, F5, 15, 15) (dual of [(2231, 15), 33393, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(572, 15618, F5, 15) (dual of [15618, 15546, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 15624, F5, 15) (dual of [15624, 15552, 16]-code), using
- 1 times truncation [i] based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- 1 times truncation [i] based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(572, 15624, F5, 15) (dual of [15624, 15552, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(572, 15618, F5, 15) (dual of [15618, 15546, 16]-code), using
- net defined by OOA [i] based on linear OOA(572, 2231, F5, 15, 15) (dual of [(2231, 15), 33393, 16]-NRT-code), using
- digital (0, 7, 6)-net over F5, using
(64, 79, 15657)-Net over F5 — Digital
Digital (64, 79, 15657)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(579, 15657, F5, 15) (dual of [15657, 15578, 16]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(578, 15655, F5, 15) (dual of [15655, 15577, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(573, 15626, F5, 15) (dual of [15626, 15553, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(549, 15626, F5, 11) (dual of [15626, 15577, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 15626 | 512−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(55, 29, F5, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,5)), using
- discarding factors / shortening the dual code based on linear OA(55, 42, F5, 3) (dual of [42, 37, 4]-code or 42-cap in PG(4,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(578, 15656, F5, 14) (dual of [15656, 15578, 15]-code), using Gilbert–Varšamov bound and bm = 578 > Vbs−1(k−1) = 36384 408908 215717 630031 528034 005005 064181 862649 063085 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(578, 15655, F5, 15) (dual of [15655, 15577, 16]-code), using
- construction X with Varšamov bound [i] based on
(64, 79, large)-Net in Base 5 — Upper bound on s
There is no (64, 79, large)-net in base 5, because
- 13 times m-reduction [i] would yield (64, 66, large)-net in base 5, but