Best Known (15, 22, s)-Nets in Base 5
(15, 22, 210)-Net over F5 — Constructive and digital
Digital (15, 22, 210)-net over F5, using
- net defined by OOA [i] based on linear OOA(522, 210, F5, 7, 7) (dual of [(210, 7), 1448, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(522, 631, F5, 7) (dual of [631, 609, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(522, 633, F5, 7) (dual of [633, 611, 8]-code), using
- construction XX applied to C1 = C([151,156]), C2 = C([153,157]), C3 = C1 + C2 = C([153,156]), and C∩ = C1 ∩ C2 = C([151,157]) [i] based on
- linear OA(517, 624, F5, 6) (dual of [624, 607, 7]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {151,152,…,156}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(517, 624, F5, 5) (dual of [624, 607, 6]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {153,154,155,156,157}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {151,152,…,157}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(513, 624, F5, 4) (dual of [624, 611, 5]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {153,154,155,156}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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 C1 = C([151,156]), C2 = C([153,157]), C3 = C1 + C2 = C([153,156]), and C∩ = C1 ∩ C2 = C([151,157]) [i] based on
- discarding factors / shortening the dual code based on linear OA(522, 633, F5, 7) (dual of [633, 611, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(522, 631, F5, 7) (dual of [631, 609, 8]-code), using
(15, 22, 559)-Net over F5 — Digital
Digital (15, 22, 559)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(522, 559, F5, 7) (dual of [559, 537, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(522, 633, F5, 7) (dual of [633, 611, 8]-code), using
- construction XX applied to C1 = C([151,156]), C2 = C([153,157]), C3 = C1 + C2 = C([153,156]), and C∩ = C1 ∩ C2 = C([151,157]) [i] based on
- linear OA(517, 624, F5, 6) (dual of [624, 607, 7]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {151,152,…,156}, and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(517, 624, F5, 5) (dual of [624, 607, 6]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {153,154,155,156,157}, and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(521, 624, F5, 7) (dual of [624, 603, 8]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {151,152,…,157}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(513, 624, F5, 4) (dual of [624, 611, 5]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {153,154,155,156}, and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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 C1 = C([151,156]), C2 = C([153,157]), C3 = C1 + C2 = C([153,156]), and C∩ = C1 ∩ C2 = C([151,157]) [i] based on
- discarding factors / shortening the dual code based on linear OA(522, 633, F5, 7) (dual of [633, 611, 8]-code), using
(15, 22, 35488)-Net in Base 5 — Upper bound on s
There is no (15, 22, 35489)-net in base 5, because
- 1 times m-reduction [i] would yield (15, 21, 35489)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 476 841728 413949 > 521 [i]