1. tUAN THANH NGUYEN Nanyang Technological University (NTU), Singapore
Linear-Time Encoding/Decoding of Irreducible Words for Codes Correcting Tandem Duplications
tUAN THANH NGUYEN
Nanyang Technological University (NTU), Singapore
Joint work with:
Yeow Meng Chee
Han Mao Kiah
Johan Chrisnata

2. Our motivation Applications that store data in living organisms
Shipman et al. (2017) : CRISPR-Cas, encoding of a digital movie into the genomes of a population of living bacteria.

3. Our motivation Duplication Errors due to the biological mutations
Deletion
Insertion A C G T A C G T
is one of the two common repeats found in the human genome (More than 50%)
Substitution G A T G A T
Duplication
Duplication
plays an important role in determining an individual’s inherited traits
is believed to be the cause of several disorders
Inversion G A T
Translocation

4. Problem Classification
The duplication length
The number of errors
1.1 Bounded
1. Fixed-length duplications
A C G A C G A G C A G C A T
1.2 Unbounded
Tandem Duplication
A C G A G C A T
Given 𝑘=3
A A C G C G A G C A G C A T T
2.1 Bounded
2. Variable-length duplications
2.2 Unbounded
We focus on the worst-case scenario !

5. Notation
Given alphabet Σ 𝑞 = 0,1,…, 𝑞−1 , an integer 𝑘≥1
012
≤𝑘-irreducible
0012
012012
𝑥=0112
≤𝑘-descendant cone of 𝑥
𝐷 ≤𝑘 (𝑥)
01122
≤𝑘-descendants of 𝑥

6. Problem Formulation Previous Works 0122 0112
Goal: Given 𝑘,𝑛,𝑞∈𝑁, construct a code 𝒞 ≤𝑘 𝑛,𝑞 such that
“For all 𝑥,𝑦 ∈ 𝒞 ≤𝑘 𝑛,𝑞 , we have 𝐷 ≤𝑘 𝑥 ∩ 𝐷 ≤𝑘 𝑦 =∅.”
Previous Works
0122
0112
Optimal codes are found when 𝑘≤2. (Jain et al. 2017)
A method to construct codes when 𝑘=3 is provided (Jain et al. 2017).
Main idea: using “irreducible words”
There is no known result when 𝑘≥4.
01122

7. For 𝑘≤3, different irreducible words generate different descendants!
Previous Work
𝑘≤2: The code is optimal
0120
0121
1201
1210 b d c a
For 𝑘≤3, different irreducible words generate different descendants!

8. Previous Work
0120
0121
1201
1210 d D b A c C a
For 𝑘=3, we can choose more than one codewords in each cone!
Irreducible words form an “almost optimal” code!

9. Our Main Results
Detailed analysis on constructed codes based on irreducible words when 𝑘=3, such codes are denoted by Ir r ≤𝑘 (𝑛,𝑞)
Provide an explicit formula to compute the size and asymptotic rate
Provide an upper bound for optimal code and hence conclude that Ir r ≤𝑘 (𝑛,𝑞) is almost optimal
Linear-time encoder for Ir r ≤𝑘 (𝑛,𝑞)
The extension of this encoder provides the first known encoder for previous constructed codes.
Publication: IEEE International Symposium on Information Theory 2018.

10. Encoder of 𝐈𝐫 𝐫 ≤𝒌 (𝒏;𝒒) for 𝒌=𝟐,𝟑 (Sketched idea)
Duplication
channel
encoder
Decoder
Error-decoder
Irr-decoder
Input
output x y y' x 𝓵
irreducible word
For 𝜖>0, to achieve encoding rates at least optimal rate − 𝜖, we only require
ℓ=Θ 1 𝜖 , 𝑚=Θ 1 𝜖
For 𝑧∈Ir r ≤𝑘 𝑚;𝑞 , we define the neighbours of 𝑧
𝑁 𝑧 ≜ 𝑧 ′ ∈Ir r ≤𝑘 𝑚;𝑞 𝑧 𝑧 ′ ∈Ir r ≤𝑘 (2𝑚;𝑞)} x
𝒙 𝟏
𝒙 𝟐 …
𝒙 𝒔
∆ ≤𝑘 𝑚;𝑞 ≜Min{ 𝑁 𝑧 𝑓𝑜𝑟 𝑧 ∈Ir r ≤𝑘 𝑚;𝑞 }
Irr
Irr
Irr y
𝒚 𝟏
𝒚 𝟐 …
𝒚 𝒔
∆ ≤𝑘 𝑚;𝑞 ≥ 𝑞 ℓ 𝓶

11. Example
𝑘 = 2, 𝑞 = 3, 𝑚 = 3, ℓ=1
20101 2 1 1
010
212
010
210
120
212
010
210
120
120

12. Special attention on 𝒒=𝟒
Recent Work:
Special attention on 𝒒=𝟒
The GC-content of a DNA string refers to the number of nucleotides that corresponds to G or C, and DNA strings with GC-content that are too high or too low are more prone to both synthesis and sequencing errors.
Many recent works use DNA strings whose GC-content are close to 50% or exactly 50%.
This is referred as “GC-balanced constraint”.
Our updated encoder:
Irreducible
Irreducible
GC-balanced
AAAA
ATACTA
ATGCTACG

13. Knuth Balancing Method Modified Knuth Method Irreducible + GC-balanced
Input
Input
A T C A T G A T
Flip
Flip
𝑡=4:
𝑡=4:
G C T G T G A T
log 𝑛
2 log 𝑛
G C T G T G A T
Output codeword
Output codeword
Redundancy: log 𝑛
(to encode the index t+ a look-up table)
Redundancy: 2 log 𝑛 +𝑂(1)
(linear-time encoding + no need a look-up table)

14. Recent Work: Design codes when 𝒌≥𝟒.
The size of our code is at least 𝑞−1 𝑞−2 … 𝑞−𝑘 𝑛 𝑘
In term of rate:
log 𝑞 (𝑞−1) + log 𝑞 𝑞−2 +…+ log 𝑞 (𝑞−𝑘) 𝑘 ≤𝑅≤ log 𝑞 (𝑞−1) 𝑞
Upper bound
Lower bound 5
0.8280
0.4936 10
0.9542
0.8701 15
0.9745
0.9311 20
0.9828
0.9547
𝐔𝐩𝐩𝐞𝐫 𝐛𝐨𝐮𝐧𝐝 𝐚𝐧𝐝 𝐋𝐨𝐰𝐞𝐫 𝐛𝐨𝐮𝐧𝐝 𝐨𝐟 𝐭𝐡𝐞 𝐜𝐨𝐝𝐞 𝐫𝐚𝐭𝐞 𝐰𝐡𝐞𝐧 𝒌=𝟒.

15. Summary Previous Works Our work Further work
Goal: “Given 𝑘,𝑛,𝑞∈𝑁, construct the largest code where each codeword is of length 𝑛 over 𝑞-ay alphabet that can correct unbounded tandem duplications of length at most 𝑘.”
Previous Works
Our work
Further work
Optimal codes when 𝑘≤2 (Jain et al. 2017)
A method to construct codes when 𝑘=3
Provide upper bound and lower bound for codes when 𝑘=3
Linear-time encoder for known TD codes (𝑘≤3)
(IEEE ISIT 2018)
Linear-time encoder for TD GC-balanced code
A method to construct codes when 𝑘≥4
Find optimal codes when
𝑘=3
Reduce the redundancy of the encoder for TD GC-balanced code
Design better codes when 𝑘≥4