Z-Channel Embedding

• Z-channel embedding is the design of encoding and decoding schemes for asymmetric binary channels where only 1→0 errors occur, ensuring zero decoding error under worst-case corruption.
• It employs constant-weight codes, list-decoding methods, and two-stage feedback strategies to manage error thresholds and optimize information rate.
• The approach leverages geometric frameworks and Plotkin-type thresholds to balance combinatorial code size limits with error-correction capabilities in practical applications.

Searching arXiv for the cited Z-channel papers to ground the article in the primary literature. Z-channel embedding denotes the design of encoding and decoding schemes that reliably store or transmit information over a binary Z-channel, an asymmetric channel in which only $1\to 0$ flips are permitted and $0\to 1$ flips are prohibited. In the adversarial formulation developed in recent work, a length-$n$ binary codeword is subjected to at most $\tau n$ asymmetric errors, and the central problem is to determine how much information can be embedded with zero decoding error under worst-case corruption (Lebedev et al., 2020, Polyanskii et al., 2021). Within this setting, the literature distinguishes non-adaptive codes, list-decodable codes, and two-stage strategies with one noiseless feedback use. These works establish threshold phenomena analogous to Plotkin-type points, derive combinatorial bounds on code size above and below those thresholds, and exhibit explicit two-stage constructions that use free-point allocations or list-resolution mechanisms to exploit the channel’s asymmetry (Lebedev et al., 2020, Polyanskii et al., 2021, Lebedev et al., 2022).

1. Z-channel model and geometric framework

The Z-channel is binary and asymmetric: a transmitted $0$ is always received as $0$, whereas a transmitted $1$ may be received as either $1$ or $0$. In the adversarial model, the encoder sends $x=(x_1,\ldots,x_n)\in\{0,1\}^n$, and the receiver observes $0\to 1$0 subject to

$0\to 1$1

The total number of asymmetric errors is bounded by $0\to 1$2, with $0\to 1$3 (Lebedev et al., 2020). Equivalent formulations in the literature describe the adversary as zeroing out up to $0\to 1$4 ones and never creating new ones (Polyanskii et al., 2021).

Several geometric notions organize code design for this channel. The quantity

$0\to 1$5

counts downward flips from $0\to 1$6 to $0\to 1$7. One asymmetric distance used in the two-stage analysis is

$0\to 1$8

with the relation

$0\to 1$9

Hence, on constant-weight codes, $n$0 coincides with the Hamming distance (Lebedev et al., 2020). A related formulation writes

$n$1

together with the Z-ball

$n$2

that is, all words obtainable from $n$3 by changing at most $n$4 ones to zeros and never changing zeros to ones (Polyanskii et al., 2021).

For list decoding, the center of an $n$5-tuple has a particularly simple structure. For codewords $n$6, the minimizer of the Chebyshev radius is the bitwise AND, equivalently the vector supported on the intersection of supports. In constant-weight codes of weight $n$7, this gives

$n$8

which makes support intersections the basic combinatorial object controlling Z-channel ambiguity (Polyanskii et al., 2021). This asymmetry-driven geometry is also what makes weight distributions and support inclusion tests central in practical decoding.

2. Decoding regimes: unique decoding, list decoding, and two-stage feedback

A non-adaptive asymmetric $n$9 code is a subset $\tau n$0 with $\tau n$1 such that every pair of distinct codewords has Z-distance at least $\tau n$2; such codes correct up to $\tau n$3 asymmetric errors in the worst-case sense (Lebedev et al., 2022). In the list-decoding formulation, a code is $\tau n$4-list-decodable with radius $\tau n$5 if every Z-ball of radius $\tau n$6 contains at most $\tau n$7 codewords (Polyanskii et al., 2021). The two notions differ operationally: unique decoding demands a singleton candidate set, while list decoding permits bounded ambiguity that can later be resolved by auxiliary information.

Two-stage embedding introduces a single noiseless feedback use at a designated time $\tau n$8, with $\tau n$9. For $0$0, the encoder transmits $0$1, depending only on the message $0$2. For $0$3, it may transmit $0$4, where $0$5 is the observed partial output (Lebedev et al., 2020). Formally, one may write the encoding rule as

$0$6

with $0$7, and require the decoder to recover the message for all patterns with at most $0$8 asymmetric errors (Lebedev et al., 2022).

The main asymptotic quantity for two-stage coding is

$0$9

where $0$0 is the maximum number of messages achievable with blocklength $0$1, at most $0$2 asymmetric errors, and one feedback moment (Lebedev et al., 2020). Positive-rate embedding means $0$3, while zero-rate means $0$4.

A distinct small-$0$5 two-stage framework for the single-error case $0$6 uses second-stage codebooks indexed by the first-stage observation and allocates free points to possible parent vertices in a directed graph of one-error transitions (Lebedev et al., 2022). This construction is combinatorial rather than asymptotic, but it addresses the same operational problem: how to exploit one feedback event to separate “no error in stage 1” from “one stage-1 error occurred” and recover the embedded payload.

3. Plotkin-type thresholds and phase transitions

For Z-channel list decoding, the literature identifies a threshold analogous to the Plotkin point. For fixed list parameter $0$7, define

$0$8

The maximizer is

$0$9

and the resulting threshold is

$1$0

For unique decoding, corresponding to $1$1, this yields $1$2 and $1$3 (Polyanskii et al., 2021). Below this threshold, the largest $1$4-list-decodable Z-channel codes have exponential size in $1$5; above it, any such code has size bounded independently of $1$6 (Polyanskii et al., 2021).

The two-stage problem has a distinct threshold because the encoder is allowed one adaptive update after feedback. The sharp transition established for two-stage encoding is

$1$7

with maximizer $1$8 and

$1$9

The theorem states

$1$0

(Lebedev et al., 2020). Thus, allowing one feedback moment shifts the positive-rate boundary from the unique-decoding Plotkin point $1$1 to an asymptotic two-stage threshold near $1$2.

The converse mechanism combines a list-decoding upper bound for first-stage constant-weight codes with bounds on the best high-error, low-rate codes available for second-stage disambiguation (Lebedev et al., 2020). A plausible implication is that two-stage feedback fundamentally changes the embedding threshold, but does not eliminate threshold behavior itself: beyond $1$3, first-stage ambiguity and second-stage error-correction limits together force zero asymptotic rate.

4. Two-stage embedding mechanisms

The asymptotic two-stage strategy of "Two-stage coding over the Z-channel" splits the block into $1$4 and $1$5, fixes a constant weight $1$6, and uses a $1$7-constant-weight code $1$8 of size $1$9 in stage 1 (Lebedev et al., 2020). The code is required to be $0$0-list-decodable for a range of list sizes up to $0$1. Because only $0$2 flips occur, if the transmitted stage-1 word has weight $0$3, then after receiving $0$4 the receiver can determine the exact number of stage-1 errors from

$0$5

Hence both parties know the remaining error budget $0$6 (Lebedev et al., 2020).

When $0$7 is small enough that the Z-ball around the stage-1 output intersects at most $0$8 codewords, the second stage uses a high-error, low-rate code of size $0$9 to resolve the candidate list. Decoding succeeds if

$x=(x_1,\ldots,x_n)\in\{0,1\}^n$0

where $x=(x_1,\ldots,x_n)\in\{0,1\}^n$1 is the supremum fraction of correctable asymmetric errors for codes of size $x=(x_1,\ldots,x_n)\in\{0,1\}^n$2 (Lebedev et al., 2020). For example, exact small-$x=(x_1,\ldots,x_n)\in\{0,1\}^n$3 values include $x=(x_1,\ldots,x_n)\in\{0,1\}^n$4, $x=(x_1,\ldots,x_n)\in\{0,1\}^n$5, $x=(x_1,\ldots,x_n)\in\{0,1\}^n$6, $x=(x_1,\ldots,x_n)\in\{0,1\}^n$7, $x=(x_1,\ldots,x_n)\in\{0,1\}^n$8, $x=(x_1,\ldots,x_n)\in\{0,1\}^n$9, and $0\to 1$00 (Lebedev et al., 2020).

When $0\to 1$01 is larger and the first-stage list may be exponentially large, the construction shortens the stage-1 code by fixing the positions of the ones observed in $0\to 1$02, derives an upper bound on the logarithmic size exponent of the shortened code, and then chooses a second-stage random constant-weight code of normalized weight $0\to 1$03 with rate

$0\to 1$04

Decoding succeeds if

$0\to 1$05

(Lebedev et al., 2020). In operational terms, stage 2 either transmits a finite disambiguation index when the first-stage list is small, or transmits fresh positive-rate information while simultaneously resolving residual ambiguity when the list is large.

The small-block construction of "Non-adaptive and two-stage coding over the Z-channel" uses a different two-stage mechanism specialized to $0\to 1$06. For each first-stage observation $0\to 1$07, a second-stage code $0\to 1$08 of length $0\to 1$09 has $0\to 1$10 codewords and $0\to 1$11 free points. Free points are strings not covered by any radius-1 ball around a codeword. The decoder checks whether the received second-stage word lies in $0\to 1$12. If it does, it declares “no stage-1 error” and decodes to a center; otherwise, it declares “stage-1 error occurred” and uses a map from free points to parent vertices (Lebedev et al., 2022). This realizes the same separation between error patterns and payload resolution, but with explicit finite-block codebooks.

5. Bounds on code size and rate

A central result for codes above the list-decoding Plotkin point is the sharp scaling law

$0\to 1$13

valid for sufficiently small $0\to 1$14 and all $0\to 1$15 (Polyanskii et al., 2021). Thus, above the threshold, code size becomes independent of blocklength and is controlled by the distance $0\to 1$16 above the threshold. For unique decoding, this yields the same $0\to 1$17 behavior above $0\to 1$18, and a concrete lower bound is

$0\to 1$19

(Polyanskii et al., 2021).

For constant-weight codes above the threshold, a double-counting argument gives an upper bound of order $0\to 1$20 per weight slice. The global $0\to 1$21 law arises by a non-uniform stratification into $0\to 1$22 weight slices near and away from the maximizing weight $0\to 1$23 (Polyanskii et al., 2021). This suggests that the dominant combinatorial difficulty above the threshold is not confined to a single weight level, but to a neighborhood of the critical weight where the constraint is tightest.

Below the Plotkin point, the maximum code size is exponential in $0\to 1$24, and the literature provides rate bounds for constant-weight codes. One upper bound has the form

$0\to 1$25

subject to

$0\to 1$26

where $0\to 1$27 is the mutual information of a binary pair $0\to 1$28 with joint distribution

$0\to 1$29

(Polyanskii et al., 2021). A lower bound from random coding with expurgation is

$0\to 1$30

where $0\to 1$31 is the confusability set of joint distributions whose $0\to 1$32-fold all-one mass is at least $0\to 1$33 (Polyanskii et al., 2021).

For the two-stage problem, the first-stage list-decoding component is governed by the normalized $0\to 1$34-radius $0\to 1$35, defined through explicit functions $0\to 1$36 and $0\to 1$37. The random coding theorem states that for any fixed $0\to 1$38, $0\to 1$39, and $0\to 1$40, there exist $0\to 1$41-constant-weight codes with rate at least $0\to 1$42 whose normalized $0\to 1$43-radius is at least $0\to 1$44, and in the regime $0\to 1$45,

$0\to 1$46

(Lebedev et al., 2020). The matching upper bound is Plotkin-type: for a $0\to 1$47-constant-weight code of size $0\to 1$48 that is $0\to 1$49-list-decodable with $0\to 1$50,

$0\to 1$51

which forces vanishing rate beyond $0\to 1$52 for every fixed $0\to 1$53 (Lebedev et al., 2020).

6. High-error low-rate codes and explicit finite-block constructions

The finite-size quantity

$0\to 1$54

plays two roles: it bounds the second stage of asymptotic two-stage schemes, and it describes the best achievable correction level when codebook size is fixed (Lebedev et al., 2020). For $0\to 1$55, $0\to 1$56 has an LP characterization in terms of a binary matrix $0\to 1$57 indexed by unordered pairs from $0\to 1$58 and $0\to 1$59 columns: $0\to 1$60 where the maximum is over nonnegative vectors $0\to 1$61 such that each entry of $0\to 1$62 is at most $0\to 1$63 (Lebedev et al., 2020). The same paper proves that for any $0\to 1$64 and $0\to 1$65, there exists $0\to 1$66 such that for all $0\to 1$67 there is a length-$0\to 1$68 code of size $0\to 1$69 correcting a fraction $0\to 1$70 of asymmetric errors.

A key side result is a Plotkin-type upper bound for asymmetric error-correcting codes: for $0\to 1$71, any binary code correcting a fraction $0\to 1$72 of asymmetric errors contains at most $0\to 1$73 codewords, and $0\to 1$74 as $0\to 1$75 (Lebedev et al., 2020). This complements the above-threshold scaling law for list decoding and shows that $0\to 1$76 remains the asymptotic unique-decoding barrier for fixed-size high-error codes.

The finite-block paper supplies explicit two-stage constructions for $0\to 1$77. One example uses $0\to 1$78, $0\to 1$79, and weight-dependent second-stage codes:

• for $0\to 1$80, a $0\to 1$81 codebook $0\to 1$82;
• for $0\to 1$83, a $0\to 1$84 codebook $0\to 1$85;
• for $0\to 1$86, a $0\to 1$87 codebook $0\to 1$88.

The weight-class feasibility inequalities

$0\to 1$89

hold, so the total message count is

$0\to 1$90

with rate

$0\to 1$91

(Lebedev et al., 2022). Another example with $0\to 1$92 achieves $0\to 1$93 and $0\to 1$94 (Lebedev et al., 2022).

For $0\to 1$95, the same paper reports that two-stage strategies exceed the cited full-feedback adaptive strategies for some small lengths:

• $0\to 1$96: two-stage $0\to 1$97 versus full-feedback $0\to 1$98;
• $0\to 1$99: two-stage $n$00 versus full-feedback $n$01;
• $n$02: two-stage $n$03 versus full-feedback $n$04;
• $n$05: two-stage $n$06 versus full-feedback $n$07 (Lebedev et al., 2022). This is a finite-block phenomenon; it should not be conflated with the asymptotic two-stage threshold near $n$08.

7. Relation to probabilistic models, implementation guidance, and open issues

The cited works are primarily combinatorial and adversarial. They differ from the probabilistic Z-channel in which each transmitted $n$09 is flipped to $n$10 independently with probability $n$11. In that stochastic model, with Bernoulli-$n$12 input,

$n$13

and the capacity is

$n$14

(Lebedev et al., 2022). Another equivalent capacity expression quoted in the literature is

$n$15

up to equivalent forms from prior work (Polyanskii et al., 2021). These stochastic capacities are not the same object as adversarial list-decoding capacity; the latter depends on worst-case support intersections and does not generally coincide with the Shannon model (Polyanskii et al., 2021).

For practical embedding over an adversarial Z-channel, the literature repeatedly emphasizes constant-weight organization and support-based decoding. In constant-weight blocks, a candidate codeword $n$16 is compatible with a received word $n$17 precisely when $n$18 and the weight loss is within the error budget (Polyanskii et al., 2021). In two-stage schemes, feedback reveals the first-stage error count through the received weight, and this determines whether the second stage should perform bounded-list disambiguation or use a positive-rate fallback code (Lebedev et al., 2020). The small-$n$19 framework refines this into a balls-versus-free-points test on the second-stage suffix (Lebedev et al., 2022).

Several limitations remain explicit in the current literature. Exact code size at the threshold $n$20 is open; a conjectured behavior is $n$21 (Polyanskii et al., 2021). Tight constants in the $n$22 regime above the threshold are not optimized (Polyanskii et al., 2021). For larger $n$23 and $n$24, the LP-based free-point search of the small-block two-stage method becomes more complex (Lebedev et al., 2022). The asymmetry of Z-distance also prevents a direct application of Delsarte-type LP methods based on shift-invariant sphere intersections, leaving alternative LP-like tools as an open direction (Polyanskii et al., 2021).

Taken together, these results define Z-channel embedding as a coding-theoretic program centered on asymmetric support geometry, threshold behavior, and adaptive disambiguation. The non-adaptive regime is controlled by the Z-channel Plotkin point, the above-threshold regime exhibits the sharp $n$25 law, and one-use feedback raises the positive-rate boundary to

$n$26

for two-stage strategies (Lebedev et al., 2020, Polyanskii et al., 2021).