Best Known (64, 64+16, s)-Nets in Base 5
(64, 64+16, 1957)-Net over F5 — Constructive and digital
Digital (64, 80, 1957)-net over F5, using
- net defined by OOA [i] based on linear OOA(580, 1957, F5, 16, 16) (dual of [(1957, 16), 31232, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(580, 15656, F5, 16) (dual of [15656, 15576, 17]-code), using
- 1 times code embedding in larger space [i] based on linear OA(579, 15655, F5, 16) (dual of [15655, 15576, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [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(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(579, 15655, F5, 16) (dual of [15655, 15576, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(580, 15656, F5, 16) (dual of [15656, 15576, 17]-code), using
(64, 64+16, 13282)-Net over F5 — Digital
Digital (64, 80, 13282)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(580, 13282, F5, 16) (dual of [13282, 13202, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(580, 15651, F5, 16) (dual of [15651, 15571, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [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(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(57, 26, F5, 4) (dual of [26, 19, 5]-code), using
- base reduction for projective spaces (embedding PG(3,25) in PG(6,5)) [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- extended Reed–Solomon code RSe(22,25) [i]
- algebraic-geometric code AG(F, Q+9P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F,7P) with degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- base reduction for projective spaces (embedding PG(3,25) in PG(6,5)) [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(580, 15651, F5, 16) (dual of [15651, 15571, 17]-code), using
(64, 64+16, large)-Net in Base 5 — Upper bound on s
There is no (64, 80, large)-net in base 5, because
- 14 times m-reduction [i] would yield (64, 66, large)-net in base 5, but