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# 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 ≅ Rk0 × F2k1

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

ix

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 13688021358596321120441153871517665762112681
(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 1283029257964003106549141684148518947611

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 1368802135859632112334115425951766596209257182
(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 22k0 + 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)

University of Bayreuth -