1 Weakly Mutually Uncorrelated Codes S. M. Hossein Tabatabaei Yazdi , Han Mao Kiah† and Olgica Milenkovic ∗ ∗ ECE Department, University of Illinois, Urbana-Champaign †SPMS, Nanyang Technological University, Singapore ∗ Abstract—We introduce the notion of weakly mutually uncorre- codesandbalanced,error-correctingWMUcodes.Abinarystring lated (WMU) sequences, motivated by applications in DNA-based is called balanced if half of its symbols are zero. On the other storagesystemsandsynchronizationprotocols.WMUsequencesare hand,aDNAstringistermedbalanceifithasa50%GCcontent, characterized by the property that no sufficiently long suffix of representing the percentage of symbols that are either G or C. one sequence is the prefix of the same or another sequence. In addition, WMU sequences used in DNA-based storage systems are Balanced DNA strands are more stable than DNA strands with requiredtohavebalancedcompositionsofsymbolsandtobeatlarge lowerorhigherGCcontentandtheyhavelowersequencingerror- 6mutual Hamming distance from each other. We present a number rates. Atthe same time,WMU codesatlargeHammingdistance 1of constructions for balanced, error-correcting WMU codes using limit the probability of erroneous codeword selection. 0Dyck paths, Knuth’s balancing principle, prefix synchronized and 2cyclic codes. Thepaperisorganizedasfollows.InSection2wereviewMU and introduce WMU codes, and derive boundson the maximum n 1. INTRODUCTION size of the latter family of codes. In addition, we outline a a J Mutually uncorrelated (MU) codes are a class of block codes constructionthatmeetstheupperbound.InSection3wedescribe 9 in which no proper prefix of one codeword is a proper suffix constructionsforerror-correctingWMUcodes,whileinSection4 2of the same or another codeword. MU codes were extensively wediscussbalancedWMUcodes.Ourmainresultsarepresented studied in the coding theory and combinatorics literature under in Section 5, where we first propose to use cyclic codes to ] Ta variety of names. Levenshtein introduced the codes in 1964 devise an efficient construction of WMU codes that are both Iunder the name ‘strongly regular codes’ [1], and suggested that balancedandhave errorcorrectingcapabilities. We then proceed s.thecodesbeusedforsynchronization.Inspiredbyapplicationsof to improve the cyclic code construction in terms of coding cdistributed sequences in frame synchronization as described by rate through decoupled constrained and error-correcting coding [ van Wijngaarden and Willink in [2], Bajic´ and Stojanovic´ [3] for binary strings. In this setting, we use Knuth’s balancing 1rediscovered mutually uncorrelated codes, and studied them technique [11] and DC-balanced codes [12]. vunder the name of ’cross-bifix-free’ codes. Constructions and 6 2. MU AND WMU CODES:DEFINITIONS, BOUNDS AND boundson the size of MU codes were also reportedin a number 7 CONSTRUCTIONS of recent contributions [4], [5]. In particular, Blackburn [5] 1 Throughout the paper we use the following notation: F 8analyzed these sequences under the name of ‘non-overlapping q denotes a finite field of order q 2. If not stated otherwise, we 0codes’, and provided a simple construction for a class of MU ≥ tacitlyassumethatq =2,andthatthecorrespondingfieldequals .codes with optimal cardinality. MU codes have also found 1 F = 0,1 . We let a= (a ,...,a ) Fn stand for a word of 0applications in DNA storage [6], [7]: In this setting, Yazdi et 2 { } 1 n ∈ q length n over F , and aj = (a ,...,a ), 1 i j n, stand 6al. [8] developed a new, random-access and rewritable DNA- q i i j ≤ ≤ ≤ 1based storage architecture based on DNA sequences endowed for a substring of a starting at position i and ending at position : j. Moreover, for two arbitrary words a Fn,b Fm we use vwith mutually uncorrelated address strings that allow selective ∈ q ∈ q iaccess to encoded DNA blocks. The addressing scheme based ab to denote a word of length n+m generated by appendingb X to the right-hand side of a. on MU codes was augmented by specialized DNA codes in [9]. r a Here, we generalize the family of MU codes by introducing A. MU Codes weakly mutually uncorrelated (WMU) codes. WMU codes are We say that a = (a ,...,a ) Fn is self uncorrelated if block codes in which no “long” prefixes of one codeword are no proper prefix of a 1matchesnits∈sufqfix, i.e., (a ,...,a ) = 1 i suffixesofthesameorothercodewords.WMUcodesdifferfrom (a ,...,a ), for all 1 i < n. One can extend th6is n i+1 n MU codes in so far that they allow short prefixes of codewords defi−nitiontomutuallyuncorrel≤atedsequencesasfollows:twonot to also appear as suffixes of codewords. This relaxation of necessarily distinct words a,b Fn are mutually uncorrelated prefix-suffix constraints was motivated in [8] for the purpose if no proper prefix of a appears∈as aqsuffix of b and vice versa. of improving code rates while allowing for increased precision Furthermore, we say that Fn is a mutually uncorrelated DNA fragment assembly and selective addressing. For more (MU) code if any two not nCec⊆essarqily distinct elements in are details regardingthe utility of WMU codes in DNA storage, the mutually uncorrelated. C interested readers are referred to the overview paper [10]. The maximumcardinality of MU codes was determinedup to We are concerned with determining bounds on the size of aconstantfactorbyBlackburn[5,Theorem8].Forcompleteness, WMUcodesandefficientWMUcodeconstructions.Weconsider we state this result below. both binary and quaternary WMU codes, the later class adapted for encoding over the four letters DNA alphabet A,T,C,G . Theorem 1. Let AMU(n,q) denote the maximum size of MU { } codes over Fn, for n 1 and q 2. Then there exist constants Our contributions include bounds on the largest size of WMU q ≥ ≥ 00. The result is constr|uCc1t|ive. C+e is a coset of , the minimum Hamming distance property 3 n k+1 3 C C Inad−dition, 2n+1 .Hence,thesizeof isboundedfrom follows immediately. |C2|≈ √2πn21 C To use the above construction to obtain balanced DNA code- below by: words, we map the elements in F to A,T,C,G via 4 { } C 4n+1 . 0 A, 1 C, ω T, ω+1 G. 3 √2π(n k+1)n21 7→ 7→ 7→ 7→ − Let a be a word of length n. Then it is straightforward to see Next, we slightly modify the aforementioned construction and that the word (a,a+1) has balanced GC content. This leads to combine it with the Prefix-Balanced Construction 1 to obtain a the simple construction described next. near-balanced k-WMU code A,T,C,G n with parameter D. For this purpose, we geneCrat∈e {accordin}g to the Balanced Corollary 2. Let C be an [n,k−1,d] cyclic code over F4 that WMU Construction 6. We set =C 0,1 n and construct by contains the all ones vector 1. Then 2 1 concatenating C1′ ⊆ {0,1}k−1 Cand C{1′′ ⊆}{0,1}n−k+1. HerCe, C1′ {(c+e,c+1+e):c∈C} isbalancedand isanear-balancedWMUcodewithparameter C1′′ is a GC balanced, k-WMU code with minimum Hamming D. It is easy to verify that is a near-balanced k-WMU DNA C distance 2d. 5 TableI SUMMARYOFTHEPROPOSEDCONSTRUCTIONSFORq=4. Code k-WMU k-WMU+Error-Correcting k-WMU+Balanced k-WMU+Error-Correcting +Balanced Rate C1 n−4kn+1 2(k−1)h(k4−dn1−)√+nm−∗Hk−h(1−md∗H21 )+nh(nd) C3 √2π(n4n−+k1+1)n21 √2π2(k−1)h(4nk−−d1√)n+−mk∗H−1h(md∗H)nd−21 Construction Construction 4 Construction 5 Construction 6 Construction 8 Note C1= 236 m∗H =n−k−2√n−k−1 C3= 236 m∗H =n−k−2√n−k−1 B. The Decoupled Binary Code Construction Let = a ...a a for 1 i m . We claim that 1 m i i is a Cbalan{ced error-co|rrec∈tinCg k-WM≤U c≤ode}over Fn. The next construction is a combination of the binary code C q Toclarifytheresult,noticethateachelementin iscreatedby Constructions in 5 and 6. concatenating m strings, where each string belongCs to Fs. Construction 8. For given integers n and k n, let m = n In addition, the words in inherit the distance and Cb0al⊆anceqd k+1 and let a, b and c be the binary compo≤nentwords. Nex−t, properties of . ThereforeC, is balanced and has minimum construct A,T,C,G n by applying the following steps: Hamming distaCn0ce at least d. C C ∈{ } (i) lEenncgothdeka u1s,inagndambiinniamryumbloHcakmcmodinegCd1ist⊆anc{e0,d1.}Lk−et1Φof k Nexlt<, fonr,awnyepsahiorwofthnaottanlecaenssdarbilny distinccatnnao,tbb∈eCidaenndticfaolr. denote th−e encoding function, so that Φ (a) . 1 Th≤isestablishesthattheconst1ructedcno−ncl+at1enatedcodeisWMU. 1 1 (ii) InvokeConstruction3 with n=m to generate∈aCn MU code Let l=is+j, where i= l and 0 j

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