The set of codewords from the universe code over Z4 with n = 4:

y ∈ [0000, 1000, 2000, 3000, 0100, 1100, 2100, 3100, 0200, 1200, 2200, 3200, 0300, 1300, 2300, 3300, 0010, 1010, 2010, 3010, 0110, 1110, 2110, 3110, 0210, 1210, 2210, 3210, 0310, 1310, 2310, 3310, 0020, 1020, 2020, 3020, 0120, 1120, 2120, 3120]

The 1st code over Z4 is Simplex alpha code:

C1 = [0000, 0123, 0202, 0321]

The 2ed code over Z4 is Preparata code:

C2 = [0000, 1111, 2222, 3333]

Cosets of the 1st code (obtained by adding 1st twenty y codewords to C1 following cosets formation methed over Z4):

C1(y) = {[0000, 0123, 0202, 0321], [1000, 1123, 1202, 1321] [2000, 2123, 2202, 2321], [3000, 3123, 3202, 3321], [0100, 0223, 0302, 0021], [1100, 1223, 1302, 1021], [2100, 2223, 2302, 2021], [3100, 3223, 3302, 3021], [0200, 0323, 0002, 0121], [1200, 1323, 1002, 1121], [2200, 2323, 2002, 2121], [3200, 3323, 3002, 3121], [0300, 0023, 0102, 0221], [1300, 1023, 1102, 1221], [2300, 2023, 2102, 2221], [3300, 3023, 3102, 3221], [0010, 0133, 0212, 0331], [1010, 1133, 1212, 1331], [2010, 2133, 2212, 2331], [3010, 3133, 3212, 3331]}

Cosets of the 2ed code(obtained by adding last twenty y codewords to C2 following cosets formation methed over Z4):

C2(y) = {[0110, 1221, 2332, 3003], [1110, 2221, 3332, 0003], [2110, 3221, 0332, 1003], [3110, 0221, 1332, 2003], [0210, 1321, 2032, 3103], [1210, 2321, 3032, 0103], [2210, 3321, 0032, 1103], [3210, 0321, 1032, 2103], [0310, 1021, 2132, 3203], [1310, 2021, 3132, 0203], [2310, 3021, 0132, 1203], [3310, 0021, 1132, 2203], [0020, 1131, 2202, 3313], [1020, 2131, 3202, 0313], [2020, 3131, 0202, 1313], [3020, 0131, 1202, 2313], [0120, 1231, 2302, 3013], [1120, 2231, 3302, 0013], [2120, 3231, 0302, 1013], [3120, 0231, 1302, 2013]}

Union of the cosets:

C1(y) ∪ C2(y) = [0000, 0123, 0202, 0321, 1000, 1123, 1202, 1321 2000, 2123, 2202, 2321, 3000, 3123, 3202, 3321, 0100, 0223, 0302, 0021, 1100, 1223, 1302, 1021, 2100, 2223, 2302, 2021, 3100, 3223, 3302, 3021, 0200, 0323, 0002, 0121, 1200, 1323, 1002, 1121, 2200, 2323, 2002, 2121, 3200, 3323, 3002, 3121, 0300, 0023, 0102, 0221, 1300, 1023, 1102, 1221, 2300, 2023, 2102, 2221, 3300, 3023, 3102, 3221, 0010, 0133, 0212, 0331, 1010, 1133, 1212, 1331, 2010, 2133, 2212, 2331, 3010, 3133, 3212, 3331, 0110, 1221, 2332, 3003, 1110, 2221, 3332, 0003, 2110, 3221, 0332, 1003, 3110, 0221, 1332, 2003, 0210, 1321, 2032, 3103, 1210, 2321, 3032, 0103, 2210, 3321, 0032, 1103, 3210, 0321, 1032, 2103, 0310, 1021, 2132, 3203, 1310, 2021, 3132, 0203, 2310, 3021, 0132, 1203, 3310, 0021, 1132, 2203, 0020, 1131, 2202, 3313, 1020, 2131, 3202, 0313, 2020, 3131, 0202, 1313, 3020, 0131, 1202, 2313, 0120, 1231, 2302, 3013, 1120, 2231, 3302, 0013, 2120, 3231, 0302, 1013, 3120, 0231, 1302, 2013]

Mapping the elements 0123 to GCAT:

[GGGG, GCAT, GAGA, GTAC, CGGG, CCAT, CAGA, CTAC, AGGG, ACAT, AAGA, ATAC, TGGG, TCAT, TAGA, TTAC , GCGG, GAAT, GTGA, GGAC , CCGG, CAAT, CTGA, CGAC , ACGG, AAAT, ATGA, AGAC , TCGG, TAAT, TTGA, TGAC , GAGG, GTAT, GGGA, GCAC , CAGG, CTAT, CGGA, CCAC , AAGG, ATAT, AGGA, ACAC , TAGG, TTAT, TGGA, TCAC , GTGG, GGAT, GCGA, GAAC , CTGG, CGAT, CCGA, CAAC , ATGG, AGAT, ACGA, AAAC , TTGG, TGAT, TCGA, TAAC , GGCG, GCTT, GACA, GTTC , CGCG, CCTT, CACA, CTTC , AGCG, ACTT, AACA, ATTC , TGCG, TCTT, TACA, TTTC, GCCG, CAAC, ATTA, TGGT , CCCG, AAAC, TTTA, GGGT , ACCG, TAAC, GTTA, CGGT , TCCG, GAAC, CTTA, AGGT , GACG, CTAC, AGTA, TCGT , CACG, ATAC, TGTA, GCGT , AACG, TTAC, GGTA, CCGT , TACG, GTAC, CGTA, ACGT , GTCG, CGAC, ACTA, TAGT , CTCG, AGAC, TCTA, GAGT , ATCG, TGAC, GCTA, CAGT , TTCG, GGAC, CCTA, AAGT , GGAG, CCTC, AAGA, TTCT , CGAG, ACTC, TAGA, GTCT , AGAG, TCTC, GAGA, CTCT , TGAG, GCTC, CAGA, ATCT , GCAG, CATC, ATGA, TGCT , CCAG, AATC, TTGA, GGCT , ACAG, TATC, GTGA, CGCT , TCAG, GATC, CTGA, AGCT]

Apply homopolymers constraint:

[GCAT, GAGA, GTAC, CAGA, CTAC, ACAT, ATAC, TCAT, TAGA, GTGA, CTGA, CGAC, ATGA, AGAC, TGAC, GTAT, GCAC, CTAT, ATAT, ACAC, TCAC, GCGA, CGAT, AGAT, ACGA, TGAT, TCGA, GACA, CGCG, CACA, AGCG, TGCG, TACA, GACG, AGTA, TCGT, CACG, TGTA, GCGT, TACG, CGTA, ACGT, GTCG, ACTA, TAGT, CTCG, TCTA, GAGT, ATCG, GCTA, CAGT, CGAG, ACTC, GTCT, AGAG, TCTC, CTCT, TGAG, GCTC, ATCT, GCAG, CATC, TGCT, ACAG, TATC, CGCT, TCAG, GATC, AGCT]

Apply GC-content and Hamming distance constraint with seed codewords CATC, TCGA for d = 4:

[ATCG, GTCT, AGAG, AGCT]