Classification of linear codes over a chain ring with 4 elements up to linear isometry

We used our program for the canonization of linear codes and a canonical augmentation approach in order to classify nonredundant linear codes

C ≤ Rⁿ with C ≅ R^k₀ × F₂^k₁

of small parameters k₀+k₁ ≤ n up to linear isometry.
The isometry classes are further distinguished by the minimum Lee distance. An entry

i^x

of the table tells you that there are x isometry classes of linear codes with minimum Lee distance i and parameters k₀, k₁ and n.

R = Z₄:

(k₀,k₁)	n=3	n=4	n=5	n=6	n=7	n=8	n=9	n=10
(1, 0)	2¹3¹4¹	2¹4²5¹	2¹4¹5¹6²	2¹4¹6²7¹8¹	2¹4¹6¹7¹8²9¹	2¹4¹6¹8²9¹10²	2¹4¹6¹8¹9¹10²11¹ 12¹	2¹4¹6¹8¹10²11¹ 12² 13¹
(2, 0)	1²2³	1³2¹³3¹4¹	1⁴2³⁰3⁵4¹⁰	1⁵2⁵⁸3¹⁰4⁴³5⁴6¹	1⁶2¹⁰⁰3¹⁴4¹⁰¹5²³6¹²	1⁷2¹⁶¹3¹⁸4¹⁹⁶5⁵⁰6⁷²7⁴8⁴	1⁸2²⁴⁵3²²4³³³5⁷⁵6²⁰²7³¹8³⁵	1⁹2³⁵⁸3²⁶4⁵²⁸5⁹⁹6⁴⁰³7⁸⁴8¹⁶²9¹⁴10¹
(1, 1)	1¹2⁶	1¹2¹¹3¹4⁴	1¹2¹⁶3²4¹³5¹	1¹2²¹3²4²⁵5⁴6⁵	1¹2²⁷3²4³⁶5⁶6¹⁹7¹8¹	1¹2³³3²4⁴⁸5⁷6³⁵7⁴8¹²	1¹2⁴⁰3²4⁶⁰5⁷6⁵²7⁷8³²9⁴10¹	1¹2⁴⁷3²4⁷⁴5⁷6⁶⁷7⁹8⁵⁸9¹⁰10¹⁵
(0, 2)	2¹4¹	2¹4²	2¹4²6¹	2¹4²6²8¹	2¹4²6²8²	2¹4²6²8³10¹	2¹4²6²8³10²12¹	2¹4²6²8³10³12²
(3, 0)	1¹	1⁵2⁴	1¹⁸2⁴⁴3¹	1⁴⁹2²⁸³3²⁷4²²	1¹²¹2¹²⁷⁵3¹⁸⁴4³⁷⁰5⁴6¹	1²⁵⁶2⁴⁷⁰⁵3⁶⁹⁹4²⁹⁷³5²³⁸6²³	1⁵¹²2¹⁵¹⁸⁶3²⁰⁰⁵4¹⁵²⁰⁰5²⁷⁶⁰6¹²¹²7⁴8¹	1⁹⁵¹2⁴⁴⁵⁰⁰3⁴⁷⁸²4⁵⁹⁹⁴⁴5¹³⁷⁰³6¹⁶⁸⁶⁶7⁷⁶²8²⁰⁴
(2, 1)	1²2¹	1⁷2¹⁶	1¹⁷2⁹⁶3⁴4⁴	1³³2³⁴⁹3³²4⁸⁵	1⁵⁸2⁹⁸⁵3¹⁰⁸4⁵⁵³5¹⁹6⁵	1⁹³2²³⁸²3²⁴⁶4²²²²5²⁴²6¹³³8¹	1¹⁴²2⁵²³¹3⁴⁷⁰4⁶⁶³⁰5⁹⁹⁸6¹⁴³⁵7³⁰8²⁵	1²⁰⁶2¹⁰⁶⁵⁴3⁷⁹⁸4¹⁶⁶⁶³5²⁵⁶⁵6⁷¹⁴¹7⁵⁵⁷8⁵⁶⁹
(1, 2)	1¹2²	1²2¹³4¹	1³2⁴⁰3²4⁹	1⁴2⁸⁶3⁶4⁵²5¹	1⁶2¹⁶²3¹¹4¹⁶⁰5⁹6¹¹	1⁷2²⁷⁶3¹⁶4³⁷⁵5³¹6⁷⁶7¹8⁴	1⁹2⁴⁴²3²⁴4⁷²⁴5⁶³6²⁹⁴7¹²8³⁶	1¹¹2⁶⁷⁵3³⁰4¹²⁶⁸5¹⁰⁴6⁷³⁸7⁵⁰8²¹⁵9³10¹
(0, 3)	2¹	2²4¹	2³4³	2⁴4⁷6¹	2⁶4¹¹6³8¹	2⁷4¹⁷6⁷8³	2⁹4²²6¹⁵8⁸	2¹¹4²⁹6²¹8¹⁹10²
(4, 0)		1¹	1⁹2⁵	1⁶³2¹¹⁵3¹	1³⁸¹2¹⁷¹⁸3⁸⁸4²⁸	1¹⁹⁵⁵2¹⁹²⁹²3²³⁴⁰4²³⁰²5²6¹	1⁸⁸⁹⁴2¹⁷⁴¹⁰⁰3²⁷⁷³⁰4⁷⁰¹²⁵5²¹⁵¹6²⁹	1³⁶⁸⁸⁰2^13585963²¹¹²⁰⁴4^11538715¹⁷⁶⁶⁵⁷6²¹¹²⁶8¹
(3, 1)		1³2¹	1²³2³²	1¹²¹2⁴⁵⁴3⁸4⁴	1⁴⁹⁹2⁴¹²⁵3²⁷³4²⁸⁷	1¹⁷²⁸2²⁷⁵⁵²3²⁹³⁹4⁷³⁷⁶5³⁵6⁴	1⁵³¹⁹2¹⁵⁰⁶⁷⁸3¹⁷⁶⁷²4⁹²⁸⁰⁴5⁴⁹⁶¹6⁶⁰⁸	1¹⁴⁹⁶¹2⁷¹⁹⁵¹⁶3⁷⁷¹¹⁵4⁷⁵⁹³³¹5¹⁰³⁷⁹³6⁵²⁵³⁹7⁸⁶8¹⁹
(2, 2)		1³2³	1¹⁶2⁵¹	1⁵⁴2⁴¹²3¹²4²²	1¹⁴⁹2²¹⁶⁸3¹⁴⁰4⁴³⁹	1³⁵⁹2⁸⁸³⁹3⁷¹⁷4⁴⁴²⁶5⁶⁴6¹⁴	1⁷⁸⁶2³⁰⁶⁴⁹3²⁵⁰⁵4²⁷⁶²⁹5¹⁶⁴⁶6⁹⁶³7¹8¹	1¹⁶⁰⁴2⁹⁵⁰⁵⁸3⁶⁹⁹⁰4¹²⁷⁹⁹³5¹²⁶⁴⁹6¹⁸⁷⁷⁰7¹⁹²8¹⁵³
(1, 3)		1¹2³	1³2²⁵4¹	1⁶2¹⁰⁴3³4²¹	1¹²2³²¹3¹⁵4¹⁵³5¹6¹	1²¹2⁸³⁴3⁴¹4⁷³⁵5¹⁷6¹⁹8¹	1³⁴2¹⁹³⁸3⁹⁴4²⁵⁷⁴5¹¹⁰6²⁷⁸7²8⁶	1⁵⁴2⁴¹⁵⁹3¹⁸¹4⁷⁴¹⁰5⁴⁰⁹6¹⁸⁶³7³⁹8¹¹⁷
(0, 4)		2¹	2³4¹	2⁶4⁵	2¹²4¹⁴6¹	2²¹4³⁷6⁴8¹	2³⁴4⁷⁷6¹⁹8⁴	2⁵⁴4¹⁴⁶6⁵⁷8¹⁹
(5, 0)			1¹	1¹⁴2⁶	1¹⁷⁹2²⁵⁰	1²²¹⁵2⁸⁴⁰⁹3²⁰¹4²⁸	1²⁵⁸⁹²2²³⁷⁶⁵⁷3²⁴⁸⁸⁶4¹¹⁵⁹²	1²⁸³⁰²⁹2^57964003^10654914^16841485¹⁸⁹⁴⁷6¹¹

For k₀+k₁&le 4: We calculated 5944677 nonisomorphic linear codes (using 17164860 calls of our algorithm). This table was produced in 420 minutes.

For the row (5,0) we started the classification of free submodules from the beginning. We calculated 12622568 nonisomorphic linear codes (using 58810334 calls of our algorithm). This table was produced in 2000 minutes.

R = F₂[X] / (X²) :

(k₀,k₁)	n=3	n=4	n=5	n=6	n=7	n=8	n=9	n=10
(1, 0)	2¹3¹4¹	2¹4² 5¹	2¹4¹ 5¹6²	2¹4¹ 6²7¹8¹	2¹4¹ 6¹ 7¹8²9¹	2¹4¹ 6¹ 8²9¹10²	2¹4¹ 6¹ 8¹9¹10²11¹ 12¹	2¹4¹ 6¹ 8¹10²11¹ 12² 13¹
(2, 0)	1²2³	1³2¹³3¹4¹	1⁴2³⁰3⁵4¹⁰	1⁵2⁵⁸3¹⁰4⁴³5⁴6¹	1⁶2¹⁰⁰3¹⁴4¹⁰¹5²³6¹²	1⁷2¹⁶¹3¹⁸4¹⁹⁶5⁵⁰6⁷²7⁴8⁴	1⁸2²⁴⁵3²²4³³³5⁷⁵6²⁰²7³¹8³⁵	1⁹2³⁵⁸3²⁶4⁵²⁸5⁹⁹6⁴⁰³7⁸⁴8¹⁶²9¹⁴10¹
(1, 1)	1¹2⁶	1¹2¹¹3¹4⁴	1¹2¹⁶3²4¹³5¹	1¹2²¹3²4²⁵5⁴6⁵	1¹2²⁷3²4³⁶5⁶6¹⁹7¹8¹	1¹2³³3²4⁴⁸5⁷6³⁵7⁴8¹²	1¹2⁴⁰3²4⁶⁰5⁷6⁵²7⁷8³²9⁴10¹	1¹2⁴⁷3²4⁷⁴5⁷6⁶⁷7⁹8⁵⁸9¹⁰10¹⁵
(0, 2)	2¹4¹	2¹4²	2¹4²6¹	2¹4²6²8¹	2¹4²6²8²	2¹4²6²8³10¹	2¹4²6²8³10²12¹	2¹4²6²8³10³12²
(3, 0)	1¹	1⁵2⁴	1¹⁸2⁴⁴3¹	1⁴⁹2²⁸³3²⁷4²²	1¹²¹2¹²⁷⁵3¹⁸⁴4³⁷¹5⁴	1²⁵⁶2⁴⁷⁰⁵3⁶⁹⁹4²⁹⁷⁵5²³⁸6²¹	1⁵¹²2¹⁵¹⁸⁶3²⁰⁰⁵4¹⁵²⁰⁷5²⁷⁶¹6¹²⁰³7⁴8²	1⁹⁵¹2⁴⁴⁵⁰⁰3⁴⁷⁸²4⁵⁹⁹⁵⁶5¹³⁷⁰⁶6¹⁶⁸⁵⁰7⁷⁶²8²⁰⁵
(2, 1)	1²2¹	1⁷2¹⁶	1¹⁷2⁹⁶3⁴4⁴	1³³2³⁴⁹3³²4⁸⁵	1⁵⁸2⁹⁸⁵3¹⁰⁸4⁵⁵³5¹⁹6⁵	1⁹³2²³⁸²3²⁴⁶4²²²²5²⁴²6¹³³8¹	1¹⁴²2⁵²³¹3⁴⁷⁰4⁶⁶³⁰5⁹⁹⁸6¹⁴³⁵7³⁰8²⁵	1²⁰⁶2¹⁰⁶⁵⁴3⁷⁹⁸4¹⁶⁶⁶³5²⁵⁶⁵6⁷¹⁴¹7⁵⁵⁷8⁵⁶⁹
(1, 2)	1¹2²	1²2¹³4¹	1³2⁴⁰3²4⁹	1⁴2⁸⁶3⁶4⁵²5¹	1⁶2¹⁶²3¹¹4¹⁶⁰5⁹6¹¹	1⁷2²⁷⁶3¹⁶4³⁷⁵5³¹6⁷⁶7¹8⁴	1⁹2⁴⁴²3²⁴4⁷²⁴5⁶³6²⁹⁴7¹²8³⁶	1¹¹2⁶⁷⁵3³⁰4¹²⁶⁸5¹⁰⁴6⁷³⁸7⁵⁰8²¹⁵9³10¹
(0, 3)	2¹	2²4¹	2³4³	2⁴4⁷6¹	2⁶4¹¹6³8¹	2⁷4¹⁷6⁷8³	2⁹4²²6¹⁵8⁸	2¹¹4²⁹6²¹8¹⁹10²
(4, 0)		1¹	1⁹2⁵	1⁶³2¹¹⁵3¹	1³⁸¹2¹⁷¹⁸3⁸⁹4²⁷	1¹⁹⁵⁵2¹⁹²⁹²3²³⁴⁴4²³⁰⁴5¹	1⁸⁸⁹⁴2¹⁷⁴¹⁰⁰3²⁷⁷⁴²4⁷⁰¹⁷⁴5²¹³⁵6²⁰	1³⁶⁸⁸⁰2^13585963²¹¹²³³4^11542595¹⁷⁶⁶⁵⁹6²⁰⁹²⁵7¹8²
(3, 1)		1³2¹	1²³2³²	1¹²¹2⁴⁵⁴3⁸4⁴	1⁴⁹⁹2⁴¹²⁵3²⁷³4²⁸⁷	1¹⁷²⁸2²⁷⁵⁵²3²⁹³⁹4⁷³⁷⁹5³⁴6²	1⁵³¹⁹2¹⁵⁰⁶⁷⁸3¹⁷⁶⁷²4⁹²⁸²⁵5⁴⁹⁶²6⁵⁸⁶	1¹⁴⁹⁶¹2⁷¹⁹⁵¹⁶3⁷⁷¹¹⁵4⁷⁵⁹⁴²⁷5¹⁰³⁸²⁹6⁵²⁴⁰³7⁸⁶8²³
(2, 2)		1³2³	1¹⁶2⁵¹	1⁵⁴2⁴¹²3¹²4²²	1¹⁴⁹2²¹⁶⁸3¹⁴⁰4⁴³⁹	1³⁵⁹2⁸⁸³⁹3⁷¹⁷4⁴⁴²⁶5⁶⁴6¹⁴	1⁷⁸⁶2³⁰⁶⁴⁹3²⁵⁰⁵4²⁷⁶²⁹5¹⁶⁴⁶6⁹⁶³7¹8¹	1¹⁶⁰⁴2⁹⁵⁰⁵⁸3⁶⁹⁹⁰4¹²⁷⁹⁹³5¹²⁶⁴⁹6¹⁸⁷⁷⁰7¹⁹²8¹⁵³
(1, 3)		1¹2³	1³2²⁵4¹	1⁶2¹⁰⁴3³4²¹	1¹²2³²¹3¹⁵4¹⁵³5¹6¹	1²¹2⁸³⁴3⁴¹4⁷³⁵5¹⁷6¹⁹8¹	1³⁴2¹⁹³⁸3⁹⁴4²⁵⁷⁴5¹¹⁰6²⁷⁸7²8⁶	1⁵⁴2⁴¹⁵⁹3¹⁸¹4⁷⁴¹⁰5⁴⁰⁹6¹⁸⁶³7³⁹8¹¹⁷
(0, 4)		2¹	2³4¹	2⁶4⁵	2¹²4¹⁴6¹	2²¹4³⁷6⁴8¹	2³⁴4⁷⁷6¹⁹8⁴	2⁵⁴4¹⁴⁶6⁵⁷8¹⁹

We calculated 5944937 nonisomorphic linear codes (using 17134160 calls of our algorithm). This table was produced in 380 minutes.

The Gray images

Using the Gray map we get binary blockcodes with 2^{2k₀ + k₁} Elements of length n' = 2*n. The following tables shows the minimum Hamming distance of these codes (compared to the best known linear binary codes with the same parameters).
The following tables are not complete for those k, where k = 2*k₀ + k₁ and (k₀ , k₁) is missing in the table above.

R = Z₄:

k	n'=6	n'=8	n'=10	n'=12	n'=14	n'=16	n'=18	n'=20
2	4(4)	5(5)	6(6)	8(8)	9(9)	10(10)	12(12)	13(13)
3	2(3)	4(4)	5(5)	6(6)	8(8)	8(8)	10(10)	10(11)
4	2(2)	4(4)	4(4)	6(6)	6(7)	8(8)	8(8)	10(10)
5	2(2)	2(2)	4(4)	4(4)	6(6)	8(8)	8(8)	8(9)
6	1(1)	2(2)	3(3)	4(4)	6(5)	6(6)	8(8)	8(8)
7		2(2)	2(2)	4(4)	4(4)	6(6)	6(7)	8(8)
8		1(1)	2(2)	3(3)	4(4)	6(5)	6(6)	8(8)
10			1(1)	2(2)	2(3)	4(4)	4(4)	6(6)

R = F₂[X] / (X²) :

k	n'=6	n'=8	n'=10	n'=12	n'=14	n'=16	n'=18	n'=20
2	4(4)	5(5)	6(6)	8(8)	9(9)	10(10)	12(12)	13(13)
3	2(3)	4(4)	5(5)	6(6)	8(8)	8(8)	10(10)	10(11)
4	2(2)	4(4)	4(4)	6(6)	6(7)	8(8)	8(8)	10(10)
5	2(2)	2(2)	4(4)	4(4)	6(6)	8(8)	8(8)	8(9)
6	1(1)	2(2)	3(3)	4(4)	5(5)	6(6)	8(8)	8(8)
7		2(2)	2(2)	4(4)	4(4)	6(6)	6(7)	8(8)
8		1(1)	2(2)	3(3)	4(4)	5(5)	6(6)	8(8)

Classification of linear codes over a chain ring with 4 elements up to linear isometry

R = Z4:

R = F2[X] / (X2) :

The Gray images

R = Z4:

R = F2[X] / (X2) :

R = Z₄:

R = F₂[X] / (X²) :

R = Z₄:

R = F₂[X] / (X²) :