# Splitting Sequences for Coding and Hybrid Incremental ARQ with Fragment Retransmission

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{st}position of the erroneously received symbol, j = 0, …, m − 1. The sign of ${\epsilon}_{j}$ denotes the direction of the error: positive, 0→1, when zero is erroneously perceived as one, and negative, 1→0, when one is perceived as zero. The exponent ${}^{j}$ shows the position of the corrupted bit. The code can be extended to ${\mathbb{Z}}_{{n}_{M}}$, where ${n}_{M}$ is a general Mersenne number [9]. The main feature of the code is that its code-word can be split into the sub-words that correspond to the splitting set $\mathcal{S}$, so we propose the name splitting code. If the error correction is excluded, the code’s detection capacities are equivalent to Fletcher’s checksum error detection code [10].

## 2. Mersenne Primes and Splitting Sequences for Binary Errors Correction

Symbol Length m | Mersenne Prime p_{M} = 2^{m} − 1 | Number of Elements in Splitting Set $\left|\mathcal{S}\right|$ | Code-Word Lengths (in bits) | ||
---|---|---|---|---|---|

Reed–Solomon | Extended Hamming | Splitting | |||

2 | 3 | - | 6 | 8 | - |

3 | 7 | 1 | 21 | 32 | 24 |

5 | 31 | 3 | 155 | 512 | 460 |

7 | 127 | 9 | 889 | 8192 | 7952 |

13 | 8191 | 315 | 106,483 | 33,554,432 | 33,538,076 |

**Proof.**

**Figure 2.**(

**a**) Coding, (

**b**) syndrome forming, and (

**c**) error-correcting procedures. The coding procedure requires two additions per information symbol, and two additions, two negations, and one multiplication per sub-word. The splitting control symbols ${C}_{i1}$ and ${C}_{i2}$ are byproducts of coding procedure. Text in gray marks the changes due to the truncation and shortening described in Section 3.3 and Section 3.4. l

_{i}—length of the ith sub-word; ss—number of sub-words.

^{th}sub-word.

^{th}received symbol ${\widehat{a}}_{ik}$, located in the splitting sub-stream i, as shown in Figure 2c.

## 3. Properties and Modifications of Splitting Code

#### 3.1. Correctable Error Patterns

- (1)
- m
_{1}positive errors (0→1) followed by a single negative error (1→0) and $m-{m}_{1}-1$ zeros, ${m}_{1}=0,\dots ,m-1$; - (2)
- A chain of $m-1$ adjacent positive errors (0→1) followed by zero;
- (3)
- All inversions of patterns (1) and (2) when a positive error is substituted by negative and vice versa;
- (4)
- All circular shifts of the previous patterns (1), (2), (3).

#### 3.2. Embedded Sub-Code of the Splitting Code

^{m}), while the SpC sub-code is a $\left({2}^{m},{2}^{m}-2\right)$ code defined over GF(2

^{m}–1) = $\mathrm{GF}\left({p}_{M}\right)$. So, the proposed SpC can be regarded as a “split” version of the single-symbol correcting code: its length is multiplied as many times as there are splitting symbols, but its error-correcting capabilities are reduced from a single symbol to a single bit (operations in this paragraph are decimal).

#### 3.3. Truncated Splitting Code for General Mersenne Numbers ${n}_{M}$

_{M}= ${2}^{m}$− 1. Then, some elements of the splitting set do not have maximal order, so the splitting ${\U0001d4c8}_{i}\xb7{\epsilon}_{j}$ and/or products ${z}_{k}\xb7{\epsilon}_{j}$ are not unique.

_{M}= 63. Its factors are 3, 7, 9, and 21, and their orders are 21, 9, 7, and 3, respectively. The splitting set is $\mathcal{S}=\left\{1,3,5,7,9,11,21\right\}$, but only the elements ${\U0001d4c8}_{i}$ ∈ {1, 5, 11} have maximal additive order. The splitting code can still be formed, but with the truncated splitting set ${\mathcal{S}}_{\mathrm{T}}=\left\{1,5,11\right\}$, with a lower number of the sub-words, and a lower code rate. The set of syndrome values will not be complete, raising the possibility for error detection (Figure 2c). The corresponding ring ${\mathbb{Z}}_{{n}_{M}}$ is not a field, so the sub-codes from Section 3.2. cannot correct all possible errors within a symbol, only the weights corresponding to a single-bit error. A list of Mersenne non-prime numbers, n

_{M}, and the cardinality $\left|{\mathcal{S}}_{\mathrm{T}}\right|$ of the truncated splitting sets is given in Table 2, while the corresponding splitting elements with the maximal additive order, ${\U0001d4c8}_{i}$, $i=1,\dots ,\left|{\mathcal{S}}_{\mathrm{T}}\right|$, can be found in the patent application [18].

Symbol Length m | Mersenne Non-Prime Numbers n_{M} = 2^{m} − 1 | Number and List of Non-Trivial Prime Factors of n_{M} | Number of Elements in the Truncated Splitting Set $\left|{\mathcal{S}}_{\mathbf{T}}\right|$ | Number of Elements $\U0001d4c8\in \mathcal{S}$ with Order below the Minimal |
---|---|---|---|---|

4 | 15 | 2 (3,5) | 1 | 2 |

6 | 63 | 3 (3,3,7) | 3 | 4 |

8 | 255 | 3 (3,5,17) | 8 | 6 |

9 | 511 | 2 (7,73) | 24 | 2 |

10 | 1023 | 3 (3,11,31) | 30 | 6 |

11 | 2047 | 2 (23,89) | 88 | 2 |

12 | 4095 | 5 (3,3,5,7,13) | 72 | 22 |

14 | 16,383 | 3 (3,43,127) | 378 | 6 |

15 | 32,767 | 3 (7,31,151) | 900 | 6 |

16 | 65,535 | 4 (3,5,17,257) | 1024 | 14 |

#### 3.4. Shortened Splitting Codes and Error Detection

_{i}. It implies the changes in Equations (1), (3), (4), and (7), where the term ${2}^{m}-1$ should be substituted by ${l}_{i}+1$, and $\left|\mathcal{S}\right|$ in summations of Equations (2) and (3) should be substituted by the number of sub-words, denoted as ss. The position of the erroneous byte within the sub-word is then equal to:

#### 3.5. Splitting Code for Adjacent Error Correction

^{m}− 1, but (2

^{m}—1)/3. The code can be formed, but the maximal length of sub-words is reduced and equal to (2

^{m}−1)/3 −1.

- (5)
- Two zeros, followed by (m—2) negative errors;
- (6)
- A positive error, followed by m
_{2}negative errors, then positive error and (m—m_{2}—2) zeros, m_{2}= 0, …, m − 2; - (7)
- Negative error followed by zero and by m
_{3}negative errors, then positive error followed by (m—m_{3}—3) zeros, m_{3}= 0, …, m—3; - (8)
- All inversions of patterns (5), (6), and (7) when a positive error is substituted by a negative and vice versa;
- (9)
- All circular shifts of the previous patterns (5), (6), (7) and (8).

#### 3.6. Asymmetrically Perfect Splitting Codes

## 4. Application Example: An ARQ Procedure for Selective Fragment Retransmission of Aggregated Data

_{COR}, in Figure 6a. The remaining multiple errors are detected, and for these frames, the auxiliary check symbols (incremental redundancy) are sent within the second stage. At the receiver, the check symbols are coupled with already received fragments. The third stage is initiated for fragments with multiple errors. Such errors can be either missed (residual fragment errors, R2

_{COR}and R3

_{COR}in Figure 6) or detected. The fragments with detected errors are retransmitted in the third stage, and, if their erroneous status persists, retransmitted again following the standard ARQ procedure.

_{DET}, R2

_{DET}, R3

_{DET}in Figure 6 correspond to the residual errors of the Fletchers error-detection checksums with detection capabilities comparable to the cyclic redundancy check (CRC) codes [10]. As already stated, the splitting FEC code is based on the Fletchers checksums, so if the error correction is turned off, the code is reverted to its detection origins, and the only residual frames/fragments errors are the ones when the code falsely declares a no-error event.

**Figure 6.**Different scenarios of the proposed hybrid incremental ARQ based on splitting code: (

**a**) error correction performed at frame and at fragment stages; (

**b**) error correction performed at fragment stages only; (

**c**) no error correction performed. Frame, its incremental redundancy and its erroneous fragments are sent in stages 1, 2 and 3, respectively. The frames/fragments that are error-free, corrected or with residual errors require no further transmissions.

^{m}

^{−1}− 1 = 127 symbols, half of its theoretical maximum. The number of fragments per frame is set to the theoretical maximum (eight fragments, Table 2). The simulation is performed in the Gaussian noise environment. The distribution of frames according to errors is shown in Figure 7a. Figure 7b shows the cases of multiple error frames when errors can be either detected or missed. Fragment distribution in the case of frames with multiple errors is shown in Figure 7c. To gain a better insight into the decrease in retransmissions if error correction is turned on, the absolute number of both frame and fragment retransmissions is presented in Figure 7d. Retransmission decrease is at the cost of increased residual errors that comprise both “FALSE single error” and “FALSE error-free” cases from Figure 7b,c.

^{−5}and different pairs of fragment length and number. Heat maps cover the statistics of single-error frames, multiple-error frames, frames with detected errors (retransmitted incremental redundancy), and frames with residual errors. It can be seen that the percentage of frames with residual errors is below the percentage of frames with multiple errors detected, except when both the length and number of fragments are at their maximal values.

**Figure 8.**Heat maps of frame error events, symbol length m = 8, bit error rate p = 10

^{−5}, for different fragments (sub-words) number and length. (

**a**) Single error; (

**b**) multiple errors; (

**c**) detected errors; (

**d**) missed (residual) errors.

**Figure 9.**Heat maps of fragment error events, symbol length m = 8, bit error rate p = 10

^{−5}, for different fragments (sub-words) number and length within a frame. (

**a**) Error-free fragments; (

**b**) single error; (

**c**) detected errors; (

**d**) missed (residual) errors.

^{−5}, with frames consisting of sixteen fragments, and two types of symbols, m = 12 and m = 16. The respective fragment lengths in bits are equal, as the length of 100 symbols with m = 12 is equivalent to 75 symbols with m = 16, providing the same frame error distribution in both cases (Figure 10a). The difference between m = 12 and m = 16 cases is observed in residual errors, both for frames (Figure 10b) and fragments (Figure 10c). The residual errors for m = 16 are well below the m = 12 case, due to the increased control redundancy per the same information content. However, although the difference in absolute values exists, it is by several orders of magnitude lower than the retransmission rate and does not alter it significantly (Figure 10d).

^{−5}, and a fragment length equal to 400 symbols if m = 12, and 300 symbols if m = 16. In both cases, the fragment length is equal to 4800 bits. Again, the difference between two symbol types is only reflected in residual errors (Figure 11b,c). Again, these changes do not affect the retransmission rate.

## 5. Discussion and Conclusions

^{−5}, varies sub-words (fragments) length and number up to their maximal values which are, for m = 8, equal to 30 symbols and 8 sub-words, respectively. The frame statistic reveals that the portion of multiple-error frames that correspond to residual errors (sum of false error-free and false single-error events) is well below the portion of multiple error frames that correspond to detected errors that initiate the second stage of transmission. The exception is maximal-length frames for which most, or in the case of Mersenne primes, all syndrome values are reserved for correctable errors and very little remains for detection (Figure 8c,d). On the other hand, heat maps in Figure 9c,d reveal that the percentage of fragments with detected errors exceeds the number of fragments with residual errors for all sets of parameters.

## 6. Patents

- D. Bajić, National patent no. 54806, “Low power error control code for aggregated packets of unequal length”, pp 8–14 (for Table 1 and Table 2) and pp 14–17 (for Table 3). http://pub.zis.gov.rs/rs-pubserver/document?iDocId=90253&iepatch=.pdf, 1 August 2016.
- D. Bajić, National patent no. 54807, “Hybrid error-control procedure with selective retransmission for aggregated unequal length packet transmission using low power integer code“, 1 August 2016.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations and Notations

Abbreviations | |

AFR | Aggregation with fragment retransmission; |

ARQ | Automatic repeat request; |

CC-HARQ | Chase combining hybrid automatic repeat request; |

IR-HARQ | Incremental redundancy hybrid automatic repeat request; |

CRC | Cyclic redundancy check; |

FEC | Forward error control; |

SpC | Splitting code; |

QoS | Quality of Service; |

RS code | Reed–Solomon code; |

WOM | Write-once memory; |

Notations | |

$\mathcal{G},\u210a\in \mathcal{G}$ | Abelian group, element of Abelian group; |

$\mathcal{E},{\epsilon}_{j}\in \mathcal{E},j$ | Multiplier set that corresponds to a single bit error weight, its element, and the corresponding index; $\mathcal{E}=\left\{\pm {2}^{0},\text{}\pm {2}^{1},\dots ,\text{}\pm {2}^{m-1}\right\};$ |

${\mathcal{E}}_{2}$ | Multiplier set that corresponds to a single bit and adjacent bits within a symbol, including circular adjacency; ${\mathcal{E}}_{2}=\left\{\pm {2}^{0},\text{}\dots ,\text{}\pm {2}^{m-1},\pm 3\xb7{2}^{0},\dots ,\pm 3\xb7{2}^{m-1}\right\};$ |

$\mathcal{S},{\U0001d4c8}_{i}\in \mathcal{S}$, $i$ | Splitting set, its element and the corresponding index (Table 1); |

${\mathcal{S}}_{\mathrm{T}}$ | Truncated splitting set for Mersenne non-prime numbers (Table 1); |

${\mathcal{S}}_{\mathrm{T}2}$ | Truncated splitting set for adjacent error correction (Table 3); |

m | Number of bits in symbol; |

${n}_{M},{p}_{M}$ | Mersenne number ${2}^{m}$−1, Mersenne prime; |

${\mathbb{Z}}_{{p}_{M}},{z}_{k}\in \text{}{\mathbb{Z}}_{{p}_{M}},k$ | Integer ring with ${p}_{M}$ elements, ring element, the corresponding index; |

GF(${p}_{M}$) | Galois field with ${p}_{M}$ elements; |

${C}_{i1},{C}_{i2},{C}_{1},{C}_{2}$ | Auxiliary splitting control symbols of the $i$^{th} sub-word, code-word control symbols; |

${S}_{i1},{S}_{i2},{S}_{1},{S}_{2}$ | Syndromes of the $i$^{th} sub-word, code-word syndromes; |

${C}_{C}$ | Number of code-words in the code-word space; |

${C}_{S}$ | Number of code-words with allowed error weights; |

${C}_{EF}$ | Number of error-free code-words; |

R1_{DET}, R2_{DET}, R3_{DET} | Frames or fragments with residual errors if only error detection is turned on, at stages 1, 2 and 3; |

R1_{COR}, R2_{COR}, R3_{COR} | Frames or fragments with residual errors if error correction is also turned on, at stages 1, 2 and 3; |

${l}_{i}$ | Length of the $i$^{th} sub-word, if the sub-words are of different length; the maximal length is ${2}^{m}$−2; |

${a}_{ik}$ | The $k$^{th} information symbol in the $i$^{th} sub-word; |

ss | Number of sub-words; the maximal values are given in Table 1, Table 2 and Table 3; |

Operators | |

$\left|.\right|$ | cardinality (number of elements); |

$\in $ | is element of; |

$\wedge $ | logical and; |

$\backslash $ | set difference; |

^ | value estimated at the receiver. |

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**Figure 1.**A code-word of the splitting code, m = 5, p

_{M}= 31, and $\left|\mathcal{S}\right|$ = 3 for sub-words comprising 30 symbols each. The splitting control symbols, C

_{1i}and C

_{2i}, I = 1, 2, 3, are byproducts of control symbol formation; the sub-words and splitting control symbols marked by the same color create independent embedded sub-code-words of the splitting code.

**Figure 3.**The effect of code-word shortening on error detection: shortening the sub-words (

**a**) and reducing the number of sub-words (

**b**) decreasing the number of syndromes that indicate correctable error, thus increasing the possibility for error detection. The number of syndromes that correspond to correctable errors can reach 100% only if 2

^{m}−1 is Mersenne prime, here shown for m = 13.

**Figure 4.**Comparison of code-word lengths for RS code that corrects an entire symbol, extended Hamming code that corrects a single bit, splitting code with $\epsilon $ and Mersenne prime or Mersenne number that corrects a single bit plus corresponding error patterns, splitting code with ${\epsilon}_{2}$ that corrects a single bit, a pair of bits plus corresponding error patterns.

**Figure 5.**The number and probability of correctable error patterns per symbol with respect to total number of error patterns, and probability of all error patterns, expressed in %; (

**a**) symbol length m as a parameter; (

**b**) maximal code-word length as a parameter. The percentage of correctable patterns is low, but these are the most probable patterns in the case of isolated and statistically independent errors. The increase in code-word length for splitting code with ${\mathcal{E}}_{2}$ is not monotonous, as can also be seen in Figure 4.

**Figure 7.**Outcomes of different stages as a function of bit error probability p, for symbol length m = 8 and frame partitioned into eight fragments of length equal to 2

^{m}

^{−1}−1 = 127 symbols. (

**a**) Frame error events distribution; (

**b**) multiple errors event distribution per frame; (

**c**) multiple error events distribution per fragment; (

**d**) number of retransmitted frames and fragments with error correction turned on and off.

**Figure 10.**Frame and fragment error events as a function of fragment lengths; increment of 1200 bits is equal to 400 m = 12 symbols, and to 75 m = 16 symbols; (

**a**) Frame error events distribution; (

**b**) residual errors in frame; (

**c**) residual errors in fragments; (

**d**) retransmitted frames and fragments.

**Figure 11.**Frame and fragment error events as a function of number of fragments in a frame: (

**a**) frame error events distribution; (

**b**) residual errors in frame; (

**c**) residual errors in fragments; (

**d**) retransmitted frames and fragments.

**Table 3.**The cardinality of the truncated splitting sets ${\mathcal{S}}_{\mathrm{T}2}\mathrm{for}{\mathcal{E}}_{2}$.

Symbol Length m | Number of Elements in the Truncated Splitting Set $\left|{\mathcal{S}}_{\mathbf{T}2}\right|$ for ${\mathcal{E}}_{2}$ |
---|---|

4 | 1 |

5 | 1 |

6 | 1 |

7 | 4 |

8 | 4 |

9 | 10 |

10 | 15 |

11 | 37 |

12 | 24 |

13 | 133 |

14 | 189 |

15 | 389 |

16 | 512 |

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**MDPI and ACS Style**

Bajić, D.; Dimić, G.; Zogović, N. Splitting Sequences for Coding and Hybrid Incremental ARQ with Fragment Retransmission. *Mathematics* **2021**, *9*, 2620.
https://doi.org/10.3390/math9202620

**AMA Style**

Bajić D, Dimić G, Zogović N. Splitting Sequences for Coding and Hybrid Incremental ARQ with Fragment Retransmission. *Mathematics*. 2021; 9(20):2620.
https://doi.org/10.3390/math9202620

**Chicago/Turabian Style**

Bajić, Dragana, Goran Dimić, and Nikola Zogović. 2021. "Splitting Sequences for Coding and Hybrid Incremental ARQ with Fragment Retransmission" *Mathematics* 9, no. 20: 2620.
https://doi.org/10.3390/math9202620