Beyond the Hamming Cliff: Compact Spatial Encodings Preserving Semantic Distance in Spiking Neural Networks

Sergio Davies; Muhammad Aitsam; Morgan-Tyler Stone; Alessandro Di Nuovo; Liangxiu Han

cs.NEOct 15, 2026

Beyond the Hamming Cliff: Compact Spatial Encodings Preserving Semantic Distance in Spiking Neural Networks

Authors: Sergio Davies, Muhammad Aitsam, Morgan-Tyler Stone, Alessandro Di Nuovo, Liangxiu Han

Organizations: Department of Computing and Mathematics, Manchester Metropolitan University, John Dalton Building, Oxford Road, Manchester, M15 6BH, United Kingdom · Centre of Excellence in Artificial Intelligence and Robotics, Sheffield Hallam University, Howard Street, Sheffield, S1 1WB, United Kingdom · Università degli Studi di Catania, Viale Andrea Doria 6, Catania, 95100, Italy

Abstract

By pre-processing input signals with Gray, Thermometer, Johnson and hybrid encodings before passing them to spiking neural networks running on SpiNNaker hardware, Davies et al.(2025) address the semantic-spatial disparity that causes the ``Hamming cliff'' in standard binary representations. Continuous semantic patterns are presented as a set of simultaneous spikes at the input layer, preserving ultra-low latency processing. Training the network on multiple continuous values under each scheme, and measuring accuracy, required homeostatic scaling and false-positive rates, shows that removing the cliff is necessary but not sufficient: Gray encoding removes it yet retains little global distance structure, whereas Thermometer and Johnson preserve semantic distance at a code width that grows with the value range. The proposed hybrid encodings contain all three as limiting cases of a single parameter, recovering most of that structure at a fraction of the width. The homeostatic factor is shown to set a radius of tolerance around each trained pattern, so that the choice of encoding alone decides which values the network treats as similar.

Figures & tables

Representation	Decision available	Input events per value	How code distance relates to $d$
Rate	after an integration window	many, growing with precision	graded, in spike counts
Latency (time to first spike)	up to $T$ , depending on the value	one per input neuron	graded, in spike times
Rank order	once enough of the order has arrived	one per neuron of the population	relative order across neurons, not the magnitude of one value
Population, Gaussian receptive fields	usually up to $T$ , as fields are read out in time	several, one per active field	preserved locally; binarised to a window of $k$ active neurons, $H=2\min(d,k)$ , saturating beyond the field width
One-hot (place)	one timestep	$N$	none: every distinct pair at $H=2$
Binary	one timestep	$\lceil\log_{2}N\rceil$	none, with cliffs at power-of-two boundaries

Table 1: Low-latency input representations compared on the three properties that distinguish a spatial code. Event counts are for a scalar of

N

levels; for the spatial codes every bit fires on one rail of its pair, so a code of width

W

delivers

W

events whatever its content.

T

is the length of the encoding window,

d

the semantic distance

d_{S}

, and

k

the width of a binarised receptive field.

Parameter	Symbol	Value
Membrane time constant	$\tau_{m}$	$20$ ms
Membrane capacitance	$C_{m}$	$1.0$ nF
Resting potential	$V_{\text{rest}}$	$-65$ mV
Reset potential	$V_{\text{reset}}$	$-65$ mV
Firing threshold	$V_{\text{th}}$	$-50$ mV a
Refractory period	$\tau_{\text{refrac}}$	$0.1$ ms

Table 2: Neuron, synapse and simulation parameters. Neuron values are the sPyNNaker defaults for IF_curr_delta , as logged on the hardware.

Figure 1: The network used for training. The pattern generator’s two neurons fire at the times encoding a “1” and a “0” respectively, and the code word of the presented value is realised as the connectivity between them and the dual-rail encoding populations (Section 3.2 ). In blue the STDP-enabled synapses, whose weights begin at zero; in orange the save neuron, which triggers the deferred weight updates; in red the teacher neuron, the only source of drive to the output neuron during training. Each encoding has its own pair of encoding populations and its own output neuron, and all eight share the generator, save and teacher neurons.

Figure 2: The network used for testing. The teacher and save neurons are removed and the learned weights are applied as static synapses, wired reciprocally so that a bit on which two trained code words disagree contributes equal excitation and inhibition and cancels exactly (Eq. 6 ). In red the excitatory synapses, in blue the inhibitory ones.

Metric	Formula	What it measures
Accuracy	$\dfrac{t_{p}+t_{n}}{t_{p}+t_{n}+f_{p}+f_{n}}$	the share of all values classified correctly; dominated by $t_{n}$ when few values are trained
Precision	$\dfrac{t_{p}}{t_{p}+f_{p}}$	the share of accepted values that were trained
Negative prediction	$\dfrac{t_{n}}{t_{n}+f_{n}}$	the share of rejected values that were untrained
Sensitivity	$\dfrac{t_{p}}{t_{p}+f_{n}}$	the share of trained values that the network detects
Specificity	$\dfrac{t_{n}}{t_{n}+f_{p}}$	the share of untrained values that the network rejects

Table 3: The five classification metrics used in Davies et al. (2025) , evaluated for every experiment and tabulated in full in Appendix A .

Encoding	Width	$127$	$128$	$H$
Binary	8	01111111	10000000	8
Gray	8	01000000	11000000	1
Thermometer	255	$1^{127}0^{128}$	$1^{128}0^{127}$	1
Hybrid lin. $B{=}4$	65	$1^{31}0^{34}$	$1^{32}0^{33}$	1
Hybrid lin. $B{=}8$	34	$1^{15}0^{19}$	$1^{16}0^{18}$	1
Johnson	128	$1^{127}0^{1}$	$1^{128}$	1

Table 4: Code words of the canonical Hamming cliff pair

127

and

128

. Long codes are written in run-length form,

1^{a}0^{b}

denoting

a

ones followed by

b

zeros.

$v$	$b$	$r$	refl.	coarse	fine	$\Delta$
0	0	0	0	000	00	—
1	0	1	1	000	01	1
2	0	2	2	000	11	1
3	0	3	3	000	10	1
4	1	0	3	100	10	1
5	1	1	2	100	11	1

Table 5: The linear hybrid construction at

N=16

and

B=4

, giving a

5

-bit code: three coarse bits for the four blocks and two fine bits for the offset. For each value

v

b=\lfloor v/B\rfloor

is the block it falls in and

r=v\bmod B

its offset within that block; “refl.” is the offset after reflection, equal to

r

in even blocks and to

B-1-r

in odd ones. “Coarse” is the Thermometer code of

b

and “fine” the Gray code of the reflected offset; the two concatenated form the code word. Because of the reflection the fine field runs

00,01,11,10

through block

0

and back again through block

1

, so that

\Delta

, the number of bits that change from the previous value, is one everywhere, including at the block boundaries

v=4

8

and

12

Panel of Figure 3 — Figure 3: Hamming distance from the anchor value $240$ to every value in $0$ – $255$ , under each of the eight encodings. Top row, the established codes; bottom row, the hybrids derived from them. Vertical scales differ between panels, since the code widths range from $8$ to $255$ bits: the comparison is of shape, not of magnitude.

Panel of Figure 4 — Figure 4: Hamming distance $H(f(i),f(j))$ between every pair of values in $0$ – $255$ , under each of the eight encodings. Thermometer is a plane rising away from the diagonal, since $H=|i-j|$ exactly; Johnson folds that plane into a ridge, the distance falling again beyond $N/2$ as the values meet around the ring. The hybrids reproduce those shapes in stepped form, the step width being the block size. Binary and Gray are flat and textured: distance is roughly constant away from the diagonal and carries little information about separation. Vertical scales differ between panels, since the code widths range from $8$ to $255$ bits.

	Linear		Cyclic
$B$	width	$\rho_{s}$	width	$\rho_{s}$	equivalent encoding
1	255	1.000	128	1.000	Thermometer / Johnson
2	128	1.000	65	0.999
4	65	0.998	34	0.996
8	34	0.990	19	0.980
16	19	0.956	12	0.911

Table 6: The hybrid family across block sizes at

N=256

\rho_{s}

is Spearman’s correlation between Hamming and semantic distance, circular for the cyclic variant.

B=1

coincides bit for bit with Thermometer and Johnson. From

B=N/2

both variants coincide bit for bit with Gray, and their

\rho_{s}

values differ only because the cyclic variant is scored against circular distance. The cyclic variant reaches Gray’s Hamming distances one step earlier: at

B=64

its coarse field is a two-bit Johnson ring, which is the two-bit Gray code with its bits swapped.

	Radius, fraction of width
Encoding	$2\%$	$5\%$	$10\%$	$20\%$
Binary	7.5	7.5	7.5	23.9
Gray	7.0	7.0	7.0	21.1
Thermometer	1.0	1.0	1.0	1.0
Hybrid lin. $B{=}4$	3.0	3.9	3.7	3.4
Hybrid lin. $B{=}8$	3.9	6.8	7.6	6.7

Table 7: Neighbourhood fragmentation at four radii: the mean number of disjoint runs of consecutive values in the Hamming ball around each anchor, averaged over all

256

anchors. The radius is a fraction of the code width, with a floor of one bit; for the

8

-bit codes the three smaller fractions all fall below one bit, so their columns coincide. The text quotes the

10\%

column.

		Hamming distance from the anchor $240$
Second	Semantic			Linear family			Cyclic family
value $v_{2}$	separation	Binary	Gray	Thermometer	$B{=}4$	$B{=}8$	Johnson	$B{=}4$	$B{=}8$
anchor $240$		$1^{4}0^{4}$	$10^{3}10^{3}$	$1^{240}0^{15}$	$1^{60}0^{5}$	$1^{30}0^{4}$	$0^{112}1^{16}$	$0^{28}1^{4}0^{2}$	$0^{14}1^{2}0^{3}$
241	1	1	1	1	1	1	1	1	1
242	2	1	2	2	2	2	2	2	2
244	4	1	2	4	2	2	4	2	2

Table 8: The fourteen second values of Section 4.2 and their Hamming distance from the anchor

240

under each encoding, with their semantic separation

|v_{2}-240|

for comparison. The groups are the close neighbours above the anchor, those below, progressively distant values, and the top of the range. Thermometer reproduces the separation exactly and Johnson its circular equivalent; the hybrids scale it by

1/B

; Binary and Gray bear no consistent relation to it. The code words of every value are listed in Table A.1 .

Encoding	Width	$f_{p}=H-1$	excess	total $f_{p}$	mean acc.
Binary	8	5/14	+26	93	0.9741
Gray	8	3/14	+6	29	0.9919
Thermometer	255	10/14	+2	630	0.8242
Hybrid lin. $B{=}4$	65	10/14	+1	155	0.9568
Hybrid lin. $B{=}8$	34	8/14	+1	81	0.9774
Johnson	128	14/14	+0	303	0.9155

Table 9: Two trained values: the anchor

240

together with each of the fourteen second values of Section 4.2 and Table 8 . If the network accepted exactly the code words lying between the two trained ones, the number of false positives would be

H-1

, where

H

is the Hamming distance between them. “

f_{p}=H-1

” counts how many of the fourteen pairs meet this exactly; “excess” is the largest number of false positives above

H-1

for any pair; “total

f_{p}

” and “mean acc.” are summed and averaged over the fourteen pairs.

Encoding	$n{=}1$	$n{=}2$	$n{=}3$	$n{=}4$	$n{=}6$	$n{=}8$
Homeostatic factor
Binary	5.18	6.94	6.95	3.48	2.32	1.74
Gray	5.17	2.96	2.30	1.73	1.39	1.04
Thermometer	2.47	1.24	0.84	0.63	0.43	0.32
Hybrid lin. $B{=}4$	2.15	1.09	0.74	0.56	0.38	0.29
Hybrid lin. $B{=}8$	2.11	1.09	0.75	0.57	0.39	0.29

Table 10: A contiguous cluster around 240, grown from one to eight trained values. Top: homeostatic factor. Middle: false positives over all 256 values. Bottom: the factor divided by

f(1)/n

, the value coherent summation would require –

1.00

means the summed weights reinforce perfectly, and the excess is the “don’t care” cancellation. Absolute factors are not comparable between encodings, because the output threshold is scaled with code width (Section 3.1 ): 8-bit codes sit near 5 and wide codes near 2 for that reason alone. Compare trends within a row, or the normalised values in the bottom block.

Encoding	total $f_{p}$	median dist.	max dist.	within 4
Binary	154	8.9	142	25%
Gray	–	–	–	–
Thermometer	12	1.7	3	100%
Hybrid lin. $B{=}4$	10	2.2	4	100%
Hybrid lin. $B{=}8$	–	–	–	–
Johnson	–	–	–	–

Table 11: Semantic distance of the false positives from the trained cluster, pooled over all trained-set sizes. “within 4” is the share falling within four semantic units – the network generalising over a contiguous zone, as opposed to firing on unrelated values. Encodings with no false positives are marked “–”.

Encoding	$W$	exact	one shell more	partial
Binary	8	31/32	0	1
Gray	8	31/32	0	1
Thermometer	255	20/32	12	0
Hybrid lin. $B{=}4$	65	28/32	4	0
Hybrid lin. $B{=}8$	34	32/32	0	0
Johnson	128	32/32	0	0

Table 12: Accepted sets against the predicted Hamming ball of radius

h=\lfloor W(1-1/m)/2\rfloor

, over all four targets and seven multipliers plus the minimum. “one shell more” counts cells in which the hardware accepted the next Hamming shell as well; “partial” counts cells at which the prediction falls exactly on a shell boundary.

Encoding	$\times 1.1$	$\times 1.25$	$\times 1.5$	$\times 2.0$	$\times 3.0$	$\times 5.0$	$\times 8.0$
Binary	1 (1)	1 (1)	9 (7)	37 (21)	37 (21)	93 (35)	93 (35)
Gray	1 (1)	1 (1)	9 (7)	37 (22)	37 (22)	93 (36)	93 (36)
Thermometer	25 (1)	42 (1)	58 (1)	80 (1)	101 (1)	118 (1)	127 (1)
Hybrid lin. $B{=}4$	20 (4)	36 (3)	52 (3)	76 (3)	96 (2)	116 (3)	124 (3)
Hybrid lin. $B{=}8$	6 (4)	27 (5)	44 (4)	68 (5)	92 (4)	108 (4)	116 (5)
Johnson	11 (1)	25 (1)	43 (1)	65 (1)	85 (1)	103 (1)	113 (1)

Table 13: Homeostatic tolerance, target 240. Each cell gives the number of values accepted and, in parentheses, the number of disjoint runs of consecutive values they form;

(1)

is a single semantic interval. Counts are limited by the end of the range, which lies only

15

values above the target. The multiplier scales each encoding’s own minimum factor.

Encoding	0	128	240	255
Binary	n/a	n/a	n/a	n/a
Gray	n/a	n/a	n/a	n/a
Thermometer	0	0	0	0
Hybrid lin. $B{=}4$	0	0	0	0
Hybrid lin. $B{=}8$	0	0	0	0
Johnson	32	0	17	32

Table 14: Values accepted only by going round the end of the range, at

\times 2.0

tolerance, by target. Only a cyclic encoding should accept any; the interior target

128

is a control at which none should. Binary and Gray are marked n/a: their accepted sets are scattered, so a value that happens to lie nearer by wrapping reflects a stray hit, not topology.

Encoding	Width	$n{=}1$	$n{=}2$	$n{=}4$	$n{=}8$
False positives
Binary	10	0	125	124	120
Gray	10	0	0	0	0
Hybrid lin. $B{=}8$	130	0	0	12	0
Hybrid lin. $B{=}16$	67	0	0	0	24
Hybrid cyc. $B{=}8$	67	0	0	0	16

Table 15: A contiguous cluster around

960

N=1024

. Thermometer (1023 bits) and Johnson (512 bits) exceed the default input capacity of Section 3.4 at this range; the comparison is with the only established codes that fit within it. Top: false positives over all 1024 values. Bottom: the homeostatic factor relative to coherent summation,

f(1)/n

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

			Linear family			Cyclic family
Value	Binary	Gray	Thermometer	$B{=}4$	$B{=}8$	Johnson	$B{=}4$	$B{=}8$
240	11110000	10001000	$1^{240}0^{15}$	$1^{60}0^{5}$	$1^{30}0^{4}$	$0^{112}1^{16}$	$0^{28}1^{4}0^{2}$	$0^{14}1^{2}0^{3}$
241	11110001	10001001	$1^{241}0^{14}$	$1^{60}0^{4}1$	$1^{30}0^{3}1$	$0^{113}1^{15}$	$0^{28}1^{4}01$	$0^{14}1^{2}0^{2}1$
242	11110010	10001011	$1^{242}0^{13}$	$1^{60}0^{3}1^{2}$	$1^{30}0^{2}1^{2}$	$0^{114}1^{14}$	$0^{28}1^{6}$	$0^{14}1^{2}01^{2}$
244	11110100	10001110	$1^{244}0^{11}$	$1^{61}0^{2}10$	$1^{30}01^{2}0$	$0^{116}1^{12}$	$0^{29}1^{4}0$	$0^{14}1^{4}0$
248	11111000	10000100	$1^{248}0^{7}$	$1^{62}0^{3}$	$1^{32}0^{2}$	$0^{120}1^{8}$	$0^{30}1^{2}0^{2}$	$0^{15}1^{2}0^{2}$

Appendix

Table A.1 : Code words of the anchor

240

and of the fourteen second values of Table 8 under each encoding. Long codes are written in run-length form,

1^{a}0^{b}

denoting

a

ones followed by

b

zeros; the Hamming distances of Table 8 can be read off by comparing each row with the anchor.

$v_{2}$	$H$	$S$	TP	TN	FP	FN	Acc.	Prec.	Neg. pred.	Sens.	Spec.
Binary (8 bits)
241	1	2.97	2	254	0	0	1.0000	1.0000	1.0000	1.0000	1.0000
242	1	2.97	2	254	0	0	1.0000	1.0000	1.0000	1.0000	1.0000
244	1	2.97	2	254	0	0	1.0000	1.0000	1.0000	1.0000	1.0000
248	1	2.97	2	254	0	0	1.0000	1.0000	1.0000	1.0000	1.0000
224	1	2.96	2	254	0	0	1.0000	1.0000	1.0000	1.0000	1.0000

Appendix

Table A.2 : Two trained values: complete classification metrics. Anchor

240

trained together with

v_{2}

;

H

is the Hamming distance between the two code words and

S

the homeostatic scaling factor. Counts are over all

256

values.

$n$	$S$	TP	TN	FP	FN	Acc.	Prec.	Neg. pred.	Sens.	Spec.	med. $d$
Binary (8 bits)
1	5.18	1	255	0	0	1.0000	1.0000	1.0000	1.0000	1.0000	–
2	6.94	2	224	30	0	0.8828	0.0625	1.0000	1.0000	0.8819	8.0
3	6.95	3	207	46	0	0.8203	0.0612	1.0000	1.0000	0.8182	12.5
4	3.48	4	224	28	0	0.8906	0.1250	1.0000	1.0000	0.8889	7.5
6	2.32	6	224	26	0	0.8984	0.1875	1.0000	1.0000	0.8960	7.0

Appendix

Table A.3 : Growing trained sets: complete classification metrics. A contiguous cluster of

n

values around

240

is trained;

S

is the homeostatic scaling factor. “med.

d

” is the median semantic distance of the false positives from the trained cluster.

$n$	$S$	TP	TN	FP	FN	Acc.	Prec.	Neg. pred.	Sens.	Spec.
Binary (10 bits)
1	4.14	1	1023	0	0	1.0000	1.0000	1.0000	1.0000	1.0000
2	6.95	2	897	125	0	0.8779	0.0157	1.0000	1.0000	0.8777
4	3.48	4	896	124	0	0.8789	0.0312	1.0000	1.0000	0.8784
8	1.75	8	896	120	0	0.8828	0.0625	1.0000	1.0000	0.8819
Gray (10 bits)

Appendix

Table A.4 : Extended range: complete classification metrics at

N=1024

. Thermometer and Johnson exceed the default input capacity at this range and are not evaluated.

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