Best Known (33−7, 33, s)-Nets in Base 5
(33−7, 33, 5211)-Net over F5 — Constructive and digital
Digital (26, 33, 5211)-net over F5, using
- net defined by OOA [i] based on linear OOA(533, 5211, F5, 7, 7) (dual of [(5211, 7), 36444, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(533, 15634, F5, 7) (dual of [15634, 15601, 8]-code), using
- construction XX applied to Ce(6) ⊂ Ce(5) ⊂ Ce(3) [i] based on
- linear OA(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(525, 15625, F5, 6) (dual of [15625, 15600, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(519, 15625, F5, 4) (dual of [15625, 15606, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(6) ⊂ Ce(5) ⊂ Ce(3) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(533, 15634, F5, 7) (dual of [15634, 15601, 8]-code), using
(33−7, 33, 15635)-Net over F5 — Digital
Digital (26, 33, 15635)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(533, 15635, F5, 7) (dual of [15635, 15602, 8]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(532, 15633, F5, 7) (dual of [15633, 15601, 8]-code), using
- construction X4 applied to C([0,6]) ⊂ C([1,5]) [i] based on
- linear OA(531, 15624, F5, 7) (dual of [15624, 15593, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(524, 15624, F5, 5) (dual of [15624, 15600, 6]-code), using the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- dual of repetition code with length 9 [i]
- linear OA(51, 9, F5, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to C([0,6]) ⊂ C([1,5]) [i] based on
- linear OA(532, 15634, F5, 6) (dual of [15634, 15602, 7]-code), using Gilbert–Varšamov bound and bm = 532 > Vbs−1(k−1) = 7963 193201 638366 341829 [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(532, 15633, F5, 7) (dual of [15633, 15601, 8]-code), using
- construction X with Varšamov bound [i] based on
(33−7, 33, large)-Net in Base 5 — Upper bound on s
There is no (26, 33, large)-net in base 5, because
- 5 times m-reduction [i] would yield (26, 28, large)-net in base 5, but