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< h1 id = "kmers-and-super-kmers" > Kmers and super-kmers</ h1 >
< h2 id = "kmers" > Kmers</ h2 >
< p > A < strong > kmer</ strong > is a DNA subsequence of fixed length < span class = "arithmatex" > \(k\)</ span > . Two constraints apply to < span class = "arithmatex" > \(k\)</ span > , both enforced when a command starts (an invalid value exits immediately with an error):</ p >
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< li >< span class = "arithmatex" > \(k \in [11, 31]\)</ span > : long enough to be specific, short enough to fit in a single 64-bit word at 2 bits/base (< span class = "arithmatex" > \(k \le 32\)</ span > is the hard limit; < span class = "arithmatex" > \(k < 11\)</ span > gives insufficient specificity).</ li >
< li >< span class = "arithmatex" > \(k\)</ span > < strong > is odd</ strong > : an odd-length sequence can never equal its own reverse complement, so the two orientations of any kmer are always distinct. This is required for the canonical form (see < a href = "../encoding/" > DNA encoding</ a > ) to be well defined.</ li >
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< h2 id = "super-kmers" > Super-kmers</ h2 >
< p > A < strong > super-kmer</ strong > is a maximal run of consecutive, overlapping kmers from a read that share the same canonical minimizer (see < a href = "../minimizer_selection/" > Minimizer selection</ a > ). Each kmer in the run overlaps the next by < span class = "arithmatex" > \(k-1\)</ span > nucleotides. A super-kmer is capped at 256 nucleotides; a longer run is split at that boundary.</ p >
< p > For a random minimizer of length < span class = "arithmatex" > \(m\)</ span > over kmers of length < span class = "arithmatex" > \(k\)</ span > , the expected length of a super-kmer is approximately (Golan & Shur 2025; Zheng < em > et al.</ em > 2020)< sup id = "fnref:Zheng2020-ji" >< a class = "footnote-ref" href = "#fn:Zheng2020-ji" > 1</ a ></ sup > < sup id = "fnref:Golan2025-xf" >< a class = "footnote-ref" href = "#fn:Golan2025-xf" > 2</ a ></ sup > :</ p >
< div class = "arithmatex" > \[L_{\text{nt}} \approx \frac{k-m+2}{2} + k - 1\]</ div >
< p > For < span class = "arithmatex" > \(k=31\)</ span > , < span class = "arithmatex" > \(m=13\)</ span > this is about 40 nucleotides; in practice super-kmers rarely exceed a few dozen nucleotides.</ p >
< h3 id = "canonical-super-kmers" > Canonical super-kmers</ h3 >
< p > A < strong > canonical super-kmer</ strong > is the lexicographic minimum of a super-kmer and its reverse complement. When a read and its reverse complement are both encountered, they produce super-kmers that are reverse complements of each other; both reduce to the same canonical super-kmer, so a genomic region is represented once regardless of which strand was read.</ p >
< p > Super-kmers are the unit of work used throughout construction and querying: sequences are decomposed into super-kmers first, and every downstream step (partition routing, deduplication, counting) operates on them rather than on individual kmers.</ p >
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< p > Zheng, H., Kingsford, C. & Marçais, G. (2020). < a href = "https://doi.org/10.1093/bioinformatics/btaa472" > Improved design and analysis of practical minimizers</ a > . < em > Bioinformatics (Oxford, England)</ em > , 36, i119--i127.  < a class = "footnote-backref" href = "#fnref:Zheng2020-ji" title = "Jump back to footnote 1 in the text" > ↩ </ a ></ p >
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< li id = "fn:Golan2025-xf" >
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< p > Golan, S. & Shur, A.M. (2025). < a href = "https://doi.org/10.1007/978-3-031-82670-2\_25" > Expected density of random minimizers</ a > . In: < em > Lecture notes in computer science</ em > , Lecture notes in computer science. Springer Nature Switzerland, Cham, pp. 347--360.  < a class = "footnote-backref" href = "#fnref:Golan2025-xf" title = "Jump back to footnote 2 in the text" > ↩ </ a ></ p >
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