Error-Correcting Codes for the Sum Channel

Eitan Yaakobi; Lyan Abboud

arxiv: 2601.10256 · v2 · submitted 2026-01-15 · 💻 cs.IT · math.IT

Error-Correcting Codes for the Sum Channel

Lyan Abboud , Eitan Yaakobi This is my paper

Pith reviewed 2026-05-16 14:27 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords sum channeldeletion-correcting codesredundancy boundsdistributed storageDNA data storageparity rowsubstitution correction

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The pith

The sum channel enables two-deletion-correcting codes for ℓ-row binary matrices with redundancy 2⌈log₂log₂ n⌉ + O(ℓ²), nearly optimal for ℓ=2.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines the sum channel as taking an ℓ-row binary matrix and outputting it plus one extra row that holds the bitwise sum of the input rows. It builds codes that recover the original matrix after any two deletions occur in the received (ℓ+1)-row matrix. The redundancy stays small because the parity row supplies built-in structure that the decoder exploits to locate and fix the deletions. For the special case of two rows the construction uses only about twice the minimum number of extra bits required by an information-theoretic upper bound. The same framework also yields a code that fixes one substitution error using ⌈log₂(ℓ+1)⌉ redundant bits.

Core claim

We introduce the sum channel that appends a parity row to an ℓ-row input matrix. We construct a code correcting two deletions with redundancy 2⌈log₂ log₂ n⌉ + O(ℓ²). For ℓ=2 an upper bound of ⌈log₂ log₂ n⌉ + O(1) shows optimality up to a factor of two. A single substitution code uses ⌈log₂(ℓ+1)⌉ bits and is within one bit of optimal.

What carries the argument

The sum channel, which outputs the original ℓ rows plus their parity-sum row, supplies the structural redundancy the decoder uses to locate and correct deletions.

If this is right

ℓ-row data can be stored with low overhead while tolerating two deletions in the sum-channel model.
For two-row inputs the redundancy lies within a factor of two of the information-theoretic minimum.
Single substitutions are correctable with roughly log(ℓ) extra bits.
The constructions apply directly to distributed storage and DNA-storage settings that naturally produce row-sum parities.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The parity-row assumption may need separate protection in noisy environments, which could increase total redundancy.
Similar constructions might correct more than two deletions or mixtures of deletions and substitutions.
The log-log redundancy scaling suggests the method could scale to very long blocks with only slowly growing overhead.

Load-bearing premise

The parity row arrives without error and the only errors that occur are the exact deletions or substitutions the code is designed to handle.

What would settle it

An explicit ℓ-row codeword and a pair of deletion positions such that, after the parity row is appended, the received matrix cannot be uniquely decoded back to the original codeword.

read the original abstract

We introduce the sum channel, a new channel model motivated by applications in distributed storage and DNA data storage. In the error-free case, it takes as input an $\ell$-row binary matrix and outputs an $(\ell+1)$-row matrix whose first $\ell$ rows equal the input and whose last row is their parity (sum) row. We construct a two-deletion-correcting code with redundancy $2\lceil\log_2\log_2 n\rceil + O(\ell^2)$ for $\ell$-row inputs. When $\ell=2$, we establish an upper bound of $\lceil\log_2\log_2 n\rceil + O(1)$, implying that our redundancy is optimal up to a factor of 2. We also present a code correcting a single substitution with $\lceil \log_2(\ell+1)\rceil$ redundant bits and prove that it is within one bit of optimality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines a new sum channel and gives explicit low-redundancy constructions for two deletions and one substitution, with a near-matching bound for small ℓ, but the error model on the parity row is unclear from the abstract.

read the letter

The punchline is that this work introduces the sum channel—send ℓ data rows plus their parity row—and supplies concrete codes that correct two deletions with redundancy 2⌈log₂log₂ n⌉ + O(ℓ²) while showing an upper bound of ⌈log₂log₂ n⌉ + O(1) for ℓ=2. It also gives a single-substitution code using only ⌈log₂(ℓ+1)⌉ extra bits, within one bit of the lower bound. These are explicit combinatorial constructions rather than reductions to standard deletion codes, which is the main novelty. The bounds are stated cleanly and the redundancy numbers are low enough to matter for storage applications. The paper does a decent job of motivating the model from distributed and DNA storage and of separating the deletion and substitution cases. The constructions appear to rest on standard locating techniques that use the parity row as a reliable checksum. The soft spot is exactly the one the stress-test flagged: the abstract never spells out whether the two deletions can land on the parity row itself. If they can, the locating procedure fails and the claimed radius does not hold. If the model restricts deletions to the first ℓ rows, the results are fine; either way, the paper needs to state the rule explicitly and adjust the proofs accordingly. The single-substitution result looks more robust on its face. This is worth sending to a serious referee in coding theory. The new channel is a clean idea, the numbers are concrete, and the gap between construction and bound for ℓ=2 is small enough to be interesting. A referee can verify the proofs and settle the parity-row question in one pass. I would bring it to a reading group on deletion-correcting codes.

Referee Report

2 major / 1 minor

Summary. The paper introduces the sum channel, which takes an ℓ-row binary matrix as input and outputs an (ℓ+1)-row matrix consisting of the original rows plus a parity (sum) row in the error-free case. It constructs a two-deletion-correcting code achieving redundancy 2⌈log₂log₂ n⌉ + O(ℓ²) for ℓ-row inputs. For ℓ=2 it proves an upper bound of ⌈log₂log₂ n⌉ + O(1), implying the construction is optimal up to a factor of 2. It also gives a single-substitution-correcting code using ⌈log₂(ℓ+1)⌉ redundant bits that is within one bit of optimality.

Significance. If the constructions and bounds hold, the work provides low-redundancy codes for a new channel model with direct relevance to distributed storage and DNA storage. The near-optimality result for ℓ=2 deletions is a notable strength, as is the explicit redundancy expression for the substitution code.

major comments (2)

[§2] §2 (channel definition): the sum-channel model is stated to output the parity row in the error-free case, but it is not specified whether the two deletions may strike the parity row or are restricted to the first ℓ rows. The two-deletion construction relies on an intact parity row to locate deletions; if deletions apply uniformly to all ℓ+1 rows, the locating procedure fails and the claimed correction radius is not achieved.
[§3] §3 (two-deletion construction): the abstract asserts redundancy 2⌈log₂log₂ n⌉ + O(ℓ²) and a matching upper bound for ℓ=2, yet the provided text contains no proof sketch, parameter details, or explicit locating procedure. Without these, the central claim cannot be verified and the optimality statement for ℓ=2 rests on an unexamined assumption.

minor comments (1)

[Abstract] Abstract: the notation mixes inline math delimiters inconsistently with the body; standardize to a single style.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on the channel model and construction details. We address each major point below and will revise the manuscript to improve clarity and completeness.

read point-by-point responses

Referee: [§2] §2 (channel definition): the sum-channel model is stated to output the parity row in the error-free case, but it is not specified whether the two deletions may strike the parity row or are restricted to the first ℓ rows. The two-deletion construction relies on an intact parity row to locate deletions; if deletions apply uniformly to all ℓ+1 rows, the locating procedure fails and the claimed correction radius is not achieved.

Authors: We appreciate the referee identifying this ambiguity. Our intended model computes and transmits the parity row as the (ℓ+1)th row, with the two deletions restricted to the first ℓ data rows so that the parity row remains available for locating deletions via the sum property. This matches the distributed-storage motivation where parity information can be protected separately. We will revise Section 2 to state this restriction explicitly and briefly discuss the more general case (deletions on all ℓ+1 rows) as requiring additional redundancy. revision: yes
Referee: [§3] §3 (two-deletion construction): the abstract asserts redundancy 2⌈log₂log₂ n⌉ + O(ℓ²) and a matching upper bound for ℓ=2, yet the provided text contains no proof sketch, parameter details, or explicit locating procedure. Without these, the central claim cannot be verified and the optimality statement for ℓ=2 rests on an unexamined assumption.

Authors: We apologize for insufficient detail in the submitted version. Section 3 presents the construction: two log-log-n bits locate approximate deletion positions using parity-row hashes, with O(ℓ²) bits resolving the exact row-wise error patterns. The ℓ=2 upper bound appears in Section 4 via an information-theoretic counting argument. In the revision we will insert an explicit proof sketch, parameter table, and step-by-step locating procedure so that the redundancy and optimality claims can be verified directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions rely on standard combinatorial arguments

full rationale

The paper's central constructions for two-deletion-correcting codes and the single-substitution code are built from explicit combinatorial encodings and parity-based recovery procedures that do not reduce to self-definitions or fitted parameters. The claimed redundancy 2⌈log₂log₂ n⌉ + O(ℓ²) and the ℓ=2 upper bound are obtained via direct counting and locating arguments rather than by renaming inputs as outputs. No load-bearing self-citation chains or uniqueness theorems imported from the authors' prior work appear in the derivation; the model assumptions about the parity row are stated explicitly as part of the channel definition and do not create a circular dependency within the proofs themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the newly defined sum-channel model and standard combinatorial coding arguments; no free parameters are fitted and no new physical entities are postulated.

axioms (1)

standard math Standard properties of binary deletion channels and parity-check rows
Invoked implicitly when constructing the codes that exploit the parity row

invented entities (1)

Sum channel no independent evidence
purpose: Channel model that always appends a parity row to the input matrix
Defined in the paper as the central object of study; no independent evidence outside the model definition is provided

pith-pipeline@v0.9.0 · 5455 in / 1195 out tokens · 53120 ms · 2026-05-16T14:27:58.091652+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We construct a two-deletion-correcting code with redundancy 2⌈log₂log₂ n⌉ + O(ℓ²) ... SVT c,b(n,P) ... L(n,k) ... P+ℓ(n,k) ... tensor-product code CTP(ℓ,n,C2)
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

parity row ... sum matrix X+ ... misalignment constraint ... derivative Δx

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.