Best Known (77−15, 77, s)-Nets in Base 5
(77−15, 77, 2235)-Net over F5 — Constructive and digital
Digital (62, 77, 2235)-net over F5, using
- 51 times duplication [i] based on digital (61, 76, 2235)-net over F5, using
- net defined by OOA [i] based on linear OOA(576, 2235, F5, 15, 15) (dual of [(2235, 15), 33449, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(576, 15646, F5, 15) (dual of [15646, 15570, 16]-code), using
- 1 times truncation [i] based on linear OA(577, 15647, F5, 16) (dual of [15647, 15570, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [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]
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 22, F5, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- 1 times truncation [i] based on linear OA(577, 15647, F5, 16) (dual of [15647, 15570, 17]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(576, 15646, F5, 15) (dual of [15646, 15570, 16]-code), using
- net defined by OOA [i] based on linear OOA(576, 2235, F5, 15, 15) (dual of [(2235, 15), 33449, 16]-NRT-code), using
(77−15, 77, 15652)-Net over F5 — Digital
Digital (62, 77, 15652)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(577, 15652, F5, 15) (dual of [15652, 15575, 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(54, 26, F5, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
(77−15, 77, large)-Net in Base 5 — Upper bound on s
There is no (62, 77, large)-net in base 5, because
- 13 times m-reduction [i] would yield (62, 64, large)-net in base 5, but