Best Known (51, 51+16, s)-Nets in Base 5
(51, 51+16, 393)-Net over F5 — Constructive and digital
Digital (51, 67, 393)-net over F5, using
- 52 times duplication [i] based on digital (49, 65, 393)-net over F5, using
- net defined by OOA [i] based on linear OOA(565, 393, F5, 16, 16) (dual of [(393, 16), 6223, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(565, 3144, F5, 16) (dual of [3144, 3079, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- OA 8-folding and stacking [i] based on linear OA(565, 3144, F5, 16) (dual of [3144, 3079, 17]-code), using
- net defined by OOA [i] based on linear OOA(565, 393, F5, 16, 16) (dual of [(393, 16), 6223, 17]-NRT-code), using
(51, 51+16, 2973)-Net over F5 — Digital
Digital (51, 67, 2973)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(567, 2973, F5, 16) (dual of [2973, 2906, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(567, 3148, F5, 16) (dual of [3148, 3081, 17]-code), using
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- linear OA(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(54, 21, F5, 3) (dual of [21, 17, 4]-code or 21-cap in PG(3,5)), using
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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
- construction XX applied to Ce(15) ⊂ Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(567, 3148, F5, 16) (dual of [3148, 3081, 17]-code), using
(51, 51+16, 672202)-Net in Base 5 — Upper bound on s
There is no (51, 67, 672203)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 67762 809716 564130 099555 220338 462462 103496 998625 > 567 [i]