Best Known (57, 57+16, s)-Nets in Base 7
(57, 57+16, 2104)-Net over F7 — Constructive and digital
Digital (57, 73, 2104)-net over F7, using
- 71 times duplication [i] based on digital (56, 72, 2104)-net over F7, using
- net defined by OOA [i] based on linear OOA(772, 2104, F7, 16, 16) (dual of [(2104, 16), 33592, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(772, 16832, F7, 16) (dual of [16832, 16760, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(772, 16833, F7, 16) (dual of [16833, 16761, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(76, 26, F7, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(772, 16833, F7, 16) (dual of [16833, 16761, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(772, 16832, F7, 16) (dual of [16832, 16760, 17]-code), using
- net defined by OOA [i] based on linear OOA(772, 2104, F7, 16, 16) (dual of [(2104, 16), 33592, 17]-NRT-code), using
(57, 57+16, 16839)-Net over F7 — Digital
Digital (57, 73, 16839)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(773, 16839, F7, 16) (dual of [16839, 16766, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- linear OA(766, 16807, F7, 16) (dual of [16807, 16741, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(741, 16807, F7, 10) (dual of [16807, 16766, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(77, 32, F7, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
(57, 57+16, large)-Net in Base 7 — Upper bound on s
There is no (57, 73, large)-net in base 7, because
- 14 times m-reduction [i] would yield (57, 59, large)-net in base 7, but